#7637119

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#39136	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4692 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,c^2=i,d^2=h,e^2=hi,f^2=g,b^a=bg,c^a=ch,d^a=dh,c^b=ci,e^c=ehi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7544315);