#7634917

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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	#2675	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4383 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)(3,7)(4,8)(5,6)(9,10,14,15)(11,18,19,16)(12,22,13,17)(20,23,21,24)(25,26,29,31)(27,30,33,38)(28,35,36,32)(34,40,39,37);(2,6)(7,8)(10,15)(16,18)(17,22)(23,24)(25,27)(26,30)(28,34)(29,33)(31,38)(32,37)(35,40)(36,39);(9,11)(10,16)(12,20)(13,21)(14,19)(15,18)(17,23)(22,24)(26,31)(27,33)(32,35)(34,39);(1,3,5,4)(2,7,6,8)(10,15)(11,19)(17,22)(20,21)(25,28,29,36)(26,32,31,35)(27,34,33,39)(30,37,38,40);(1,4,5,3)(2,8,6,7)(9,12,14,13)(10,17,15,22)(11,20,19,21)(16,23,18,24);(9,13,14,12)(10,17,15,22)(11,21,19,20)(16,23,18,24)

Polycyclic presentation:

2^9|a^2=h,d^2=gh,e^2=g,f^2=hi,b^a=bg,c^a=ch,d^a=dh,e^a=ehi,c^b=ci,d^c=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, 7542113);