#7917735

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

5438 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)(3,4)(5,7)(6,8)(9,10)(11,15)(12,14)(13,16)(17,18,23,25)(19,24,28,31)(20,30,26,21)(22,27,29,32);(1,3)(2,6)(4,7)(5,8)(17,19)(18,24)(20,29)(21,27)(22,26)(23,28)(25,31)(30,32);(2,7)(4,6)(9,10,12,14)(11,13,16,15)(18,25)(19,28)(20,26)(27,32);(1,4,5,6)(2,3,7,8)(9,10)(11,13)(12,14)(15,16)(17,20,23,26)(18,21,25,30)(19,29,28,22)(24,27,31,32);(1,3)(2,6)(4,7)(5,8)(9,11,12,16)(10,13,14,15)(17,21)(18,26)(19,27)(20,25)(22,24)(23,30)(28,32)(29,31);(17,22,23,29)(18,27,25,32)(19,26,28,20)(21,31,30,24)

Polycyclic presentation:

2^9|a^2=i,c^2=ghi,d^2=gi,e^2=ghi,f^2=i,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,d^c=dh,f^e=fi

Conjugacy classes & character table

95 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7824931);