#151980

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂×C₄×C₈ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂×C₄×C₈
Frattini, cosocle	C₂²×C₁₆	C₂³
socle	C₂²	C₂×#346
prime radical	C₂³	C₂×M₃₂
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₁₆ ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁

228 conjugacy classes of subgroups; only the 146 normal subgroups are shown.

normal subgroups only

This group has 228 conjugacy classes of subgroups — too many to draw. Only its 146 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)(7,21,24,42)(8,28)(9,16)(10,34)(11,18)(12,19)(13,20)(14,23,40,27)(15,32)(17,38)(22,43,26,50)(25,57,56,51)(29,60,53,39)(30,45)(31,44)(33,49)(35,47)(36,63)(37,46)(41,59,62,55)(48,52)(54,68,58,67)(61,65,64,66);(2,4)(5,7,10,25,11,26,36,58,12,27,37,59,32,53,35,64,13,24,38,56,30,22,52,54,9,23,33,55,8,29,31,61)(6,14,17,41,18,39,48,65,19,42,49,57,15,43,44,67,20,40,34,62,45,60,63,66,16,21,46,51,28,50,47,68);(5,8,9,30,13,32,12,11)(6,15,16,18,20,28,19,45)(7,22,27,29,24,26,23,53)(10,35,33,36,38,31,37,52)(14,39,42,43,40,60,21,50)(17,47,46,63,34,44,49,48)(25,58,59,64,56,54,55,61)(41,66,57,68,62,65,51,67)

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=de,d^2=i,f^2=g,g^2=h,h^2=i,b^a=bd,c^a=ci,f^a=fi,c^b=ch,e^b=ei,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 59176);