#135820

order	512
nilpotency class	4
isoclinism	isoclinic class of #2192
automorphism	?
homology	Abelianization (H₁) = C₂²×C₈ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#447
commutator	C₂×C₈	C₂²×C₈
Frattini, cosocle	C₂×C₄×C₈	C₂³
socle	C₂²	#1852
prime radical	C₂×#41	C₄⋊C₄
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×C₄² ◁ C₄×#131 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

312 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(1,2,5,3,4,6,8,7)(9,10)(11,22)(12,26)(13,20)(14,19)(15,21)(16,23)(17,24)(18,29)(25,37)(27,38)(28,32)(30,36)(31,35)(33,40)(34,39);(1,3,5,6,4,7,8,2)(9,11)(10,16)(12,27)(13,24)(14,31)(15,25)(17,32)(18,35)(19,33)(20,38)(21,34)(22,26)(23,29)(28,39)(30,40)(36,37);(9,11)(10,17)(12,23)(13,31)(14,24)(15,25)(16,32)(18,34)(19,37)(20,22)(21,35)(26,38)(27,29)(28,40)(30,39)(33,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,d^2=gi,e^2=i,f^2=h,g^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=ei,g^a=gi,d^b=dg,g^b=gi,e^c=ei

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 43016);