#7712134

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

4829 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7619330);