#7919911

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

4912 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:

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

Polycyclic presentation:

2^9|a^2=hi,c^2=ghi,d^2=h,f^2=gh,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, 7827107);