#7968405

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

5127 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=h,c^2=gh,e^2=gh,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,f^b=fi,d^c=dh,e^c=ei,f^e=fi

Conjugacy classes & character table

101 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7875601);