#7689161

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5448 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,e^2=gh,f^2=gh,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, 7596357);