#7678265

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

4713 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7585461);