#7681745

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

4700 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)(10,13)(14,16)(17,18)(19,23)(20,27)(21,22)(24,28)(25,29)(26,32)(30,31);(1,3,6,5)(2,4,7,8)(9,10)(11,14)(12,13)(15,16)(17,18,21,22)(19,23,25,29)(20,24,28,27)(26,30,31,32);(2,7)(4,8)(18,22)(19,25)(24,27)(26,31);(1,4,6,8)(2,3,7,5)(17,19)(18,23)(20,26)(21,25)(22,29)(24,30)(27,32)(28,31);(1,3,6,5)(2,4,7,8)(9,11,12,15)(10,14,13,16)(17,18)(19,23)(20,24)(21,22)(25,29)(26,30)(27,28)(31,32);(1,5,6,3)(2,8,7,4)(17,20)(18,24)(19,26)(21,28)(22,27)(23,30)(25,31)(29,32)

Polycyclic presentation:

2^9|b^2=h,d^2=hi,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, 7588941);