#6461077

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#28567	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1630 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,e^2=fi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dhi,d^c=di,e^d=ei

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6368273);