#7688108

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹¹×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5032 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=gi,c^2=g,d^2=g,e^2=i,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, 7595304);