#7905424

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

4864 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)(9,10,13,11)(12,15,16,14)(17,18,22,24)(19,20,26,28)(21,23,30,29)(25,27,32,31);(1,2)(3,7)(4,8)(5,6)(17,19)(18,20)(21,25)(22,26)(23,27)(24,28)(29,31)(30,32);(10,11)(14,15)(17,20,22,28)(18,19,24,26)(21,27,30,31)(23,25,29,32);(1,3,5,4)(2,7,6,8)(9,11,13,10)(12,15,16,14)(17,20)(18,19)(21,27)(22,28)(23,25)(24,26)(29,32)(30,31);(9,12,13,16)(10,14,11,15)(17,18)(19,20)(21,29)(22,24)(23,30)(25,31)(26,28)(27,32);(1,4,5,3)(2,8,6,7)(17,21,22,30)(18,23,24,29)(19,25,26,32)(20,27,28,31)

Polycyclic presentation:

2^9|a^2=h,c^2=i,d^2=ghi,e^2=hi,f^2=gi,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, 7812620);