#7918671

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

4996 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=h,d^2=ghi,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, 7825867);