#7712351

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

4868 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)(4,7)(10,13)(14,16)(17,18,24,28)(19,26,22,21)(20,27,23,25)(29,32,30,31);(1,2)(3,7)(4,8)(5,6)(9,10,12,13)(11,14,15,16)(17,19,24,22)(18,21,28,26)(20,30,23,29)(25,31,27,32);(2,6)(3,8)(17,20,24,23)(18,25,28,27)(19,29,22,30)(21,32,26,31);(17,21,24,26)(18,19,28,22)(20,31,23,32)(25,30,27,29);(1,3,5,8)(2,7,6,4)(9,11,12,15)(10,14,13,16)(17,22,24,19)(18,26,28,21)(20,29,23,30)(25,32,27,31);(1,4,5,7)(2,8,6,3)(17,23,24,20)(18,27,28,25)(19,29,22,30)(21,32,26,31)

Polycyclic presentation:

2^9|a^2=h,b^2=ghi,c^2=h,d^2=h,e^2=g,f^2=i,b^a=bg,c^a=ch,e^a=eh,f^a=fi,c^b=ci,d^c=dh,e^c=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619547);