#7733489

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
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4650 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=g,c^2=h,d^2=h,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, 7640685);