#7688974

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4531 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gh,c^2=h,d^2=i,e^2=hi,f^2=i,b^a=bg,c^a=ch,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

113 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7596170);