#7633307

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	C₂×#532	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4918 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^2=i,d^2=g,e^2=g,f^2=hi,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, 7540503);