#7896648

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

4631 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=h,e^2=g,f^2=g,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, 7803844);