#152835

order	512
nilpotency class	5
isoclinism	isoclinic class of #13697
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×D₁₆
commutator	C₂×C₁₆	C₂²×C₄
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₄×D₁₆
prime radical	C₂×#508	C₂²
LCS	G ▷ C₂×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ C₂×C₄×C₈ ◁ C₂×C₄×C₁₆ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

213 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 78 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6,11,19,14,21,13,44)(7,22,27,61,30,54,29,64)(8,31,34,50,26,60,36,65)(9,18,37,46,39,59,38,66)(10,17,23,56,41,57,40,58)(12,42,24,47,43,67,32,20)(15,45,49,63,51,35,53,25)(16,52,48,28,55,62,68,33);(5,7,23,36,43,33,9,25,11,27,41,8,32,52,37,45,14,30,40,34,12,28,39,63,13,29,10,26,24,62,38,35)(6,15,46,55,47,65,57,54,44,53,18,48,42,60,56,61,21,51,66,16,20,50,17,22,19,49,59,68,67,31,58,64);(2,4)(5,8,9,30,32,63,40,62,13,36,38,27,43,45,41,28,14,26,39,7,24,25,23,52,11,34,37,29,12,35,10,33)(6,16,17,53,42,64,46,50,19,48,56,15,47,22,59,60,21,55,57,49,67,61,66,65,44,68,58,51,20,54,18,31)

Polycyclic presentation:

2^9|a^2=f,b^2=dfg,c^2=d,d^2=gh,e^2=g,f^2=i,g^2=hi,h^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=egh,g^a=gh,h^a=hi,c^b=ci

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286872, #72320813, #72320849
maximal normal subgroup of	#72286872, #72320813, #72320849

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60031);