#7681341

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

4854 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)(3,4)(5,8)(6,7)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,28)(21,23)(22,26)(25,29)(27,30)(31,32);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,15)(13,16)(17,19)(18,22)(20,27)(21,26)(23,24)(25,30)(28,31)(29,32);(2,7)(4,8)(18,23)(19,26)(25,28)(27,32);(1,4,6,8)(2,3,7,5)(9,11,13,12)(10,14,16,15)(17,19)(18,24)(20,27)(21,26)(22,23)(25,31)(28,30)(29,32);(1,3,6,5)(2,4,7,8)(9,12,13,11)(10,15,16,14)(17,18)(19,24)(20,25)(21,23)(22,26)(27,31)(28,29)(30,32);(1,5,6,3)(2,8,7,4)(17,20)(18,25)(19,27)(21,29)(22,30)(23,28)(24,31)(26,32)

Polycyclic presentation:

2^9|a^2=gi,b^2=hi,d^2=gh,e^2=gh,f^2=hi,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, 7588537);