#7674601

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

4656 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)(6,7)(9,10,13,12)(11,15,16,14)(17,18,23,20)(19,26,21,24)(22,28,30,25)(27,31,29,32);(9,11,13,16)(10,14,12,15)(17,19,23,21)(18,24,20,26)(22,27,30,29)(25,31,28,32);(1,2)(3,7)(4,5)(6,8)(19,21)(24,26)(27,29)(31,32);(1,3,4,8)(2,6,5,7)(9,12,13,10)(11,15,16,14)(17,20,23,18)(19,24,21,26)(22,28,30,25)(27,31,29,32);(17,21,23,19)(18,24,20,26)(22,29,30,27)(25,31,28,32);(17,22,23,30)(18,25,20,28)(19,27,21,29)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=gh,e^2=i,f^2=i,b^a=bg,c^a=ch,f^a=fi,c^b=ci,d^b=di,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, 7581797);