#6694778

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667460
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	#24237	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄×#123 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1607 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)(3,7)(4,6)(5,8)(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,29,26)(21,24,31,30)(25,28,32,27);(1,3)(2,5)(4,8)(6,7)(17,19)(18,23)(20,26)(21,27)(22,29)(24,25)(28,31)(30,32);(10,13)(14,16)(17,18,22,20)(19,25,29,32)(21,30,31,24)(23,28,26,27);(2,6)(5,7)(9,11,12,15)(10,14,13,16)(17,20,22,18)(19,26,29,23)(21,30,31,24)(25,27,32,28);(3,8)(5,7)(19,27)(21,31)(23,25)(24,30)(26,32)(28,29)

Polycyclic presentation:

2^9|a^2=i,c^2=i,d^2=gi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,h^e=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6601974);