#7637366

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	#2856	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4688 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)(9,10,13,12)(11,14,16,15)(17,18,21,24)(19,25,27,22)(20,28,29,23)(26,30,32,31);(2,5)(6,8)(9,10,13,12)(11,15,16,14);(9,11,13,16)(10,14,12,15)(17,19)(18,22)(20,26)(21,27)(23,30)(24,25)(28,31)(29,32);(17,20,21,29)(18,23,24,28)(19,26,27,32)(22,30,25,31);(9,12,13,10)(11,14,16,15)(19,27)(22,25)(26,32)(30,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=i,c^2=i,d^2=h,e^2=i,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, 7544562);