#7733977

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

4979 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=gh,d^2=h,e^2=h,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,f^b=fi,d^c=dhi,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, 7641173);