#7752431

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

4961 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7659627);