#7643592

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

5082 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=dh,f^a=fhi,c^b=ci,d^b=dhi,e^c=ehi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7550788);