#7633806

order	512
nilpotency class	2
isoclinism	isoclinic class of #37436
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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4336 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 40 points:

(1,2)(3,6)(4,5)(7,8)(9,10,13,15)(11,14,18,22)(12,19,21,16)(17,24,20,23)(25,26)(27,30)(28,31)(29,32)(33,37)(34,38)(35,39)(36,40);(2,5)(6,8)(9,11,13,18)(10,14,15,22)(12,20,21,17)(16,24,19,23)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(10,15)(14,22)(16,19)(23,24)(25,28,29,36)(26,31,32,40)(27,33,34,35)(30,37,38,39);(9,12,13,21)(10,16,15,19)(11,17,18,20)(14,23,22,24)(27,34)(30,38)(33,35)(37,39);(10,15)(14,22)(16,19)(23,24)(26,32)(30,38)(31,40)(37,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7541002);