#7707635

order	512
nilpotency class	2
isoclinism	isoclinic class of #47826
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

4301 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)(9,10,13,16)(11,17,19,14)(12,15,21,24)(18,23,20,22)(25,26,31,34)(27,32,28,36)(29,33,30,38)(35,39,37,40);(2,5)(6,8)(25,27)(26,32)(28,31)(29,37)(30,35)(33,40)(34,36)(38,39);(9,11,13,19)(10,14,16,17)(12,20,21,18)(15,23,24,22)(25,28,31,27)(26,32,34,36)(29,37,30,35)(33,39,38,40);(9,12,13,21)(10,15,16,24)(11,18,19,20)(14,22,17,23)(25,29,31,30)(26,33,34,38)(27,35,28,37)(32,39,36,40);(10,16)(14,17)(15,24)(22,23)(25,30,31,29)(26,33,34,38)(27,35,28,37)(32,40,36,39);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=h,e^2=i,f^2=g,b^a=bg,c^a=ch,e^a=eh,c^b=ci,d^b=di,d^c=dhi,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, 7614831);