#7632265

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

5050 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

122 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7539461);