#152811

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₁₆ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂×D₄
commutator	C₂×C₄	C₂²×C₁₆
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	D₄×C₁₆
prime radical	C₂×#41	C₂×C₈
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ C₂²×C₃₂ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

213 conjugacy classes of subgroups; only the 94 normal subgroups are shown.

normal subgroups only

This group has 213 conjugacy classes of subgroups — too many to draw. Only its 94 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)(5,6,10,17,11,15,33,42,12,16,34,37,8,27,29,57,13,18,35,44,30,43,51,65,9,32,20,49,31,58,50,61)(7,19,23,46,24,47,54,67,25,41,55,64,28,38,59,62,26,48,56,68,52,45,60,66,22,14,36,39,21,40,53,63);(5,7,20,36,30,52,29,59,12,25,10,23,31,21,51,60,13,26,34,55,11,24,50,53,9,22,35,56,8,28,33,54)(6,14,37,46,43,38,61,66,16,19,44,64,58,45,42,63,18,41,49,68,15,40,57,67,32,48,17,39,27,47,65,62);(2,4)(5,8,9,11,13,31,12,30)(6,15,16,27,18,43,32,58)(7,21,22,52,26,28,25,24)(10,29,20,33,35,50,34,51)(14,38,19,45,41,40,48,47)(17,42,37,57,44,65,49,61)(23,53,36,60,56,59,55,54)(39,62,46,66,64,63,68,67)

Polycyclic presentation:

2^9|a^2=f,b^2=df,c^2=d,d^2=i,e^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ce,d^a=di,e^a=ei,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72287227
maximal normal subgroup of	#72287227

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 60007);