#7707682

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	#392	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)(3,7)(4,8)(5,6)(9,10)(11,17)(12,16)(13,14)(15,19)(18,24)(20,23)(21,22)(25,26)(27,33)(28,32)(29,34)(30,38)(31,35)(36,40)(37,39);(2,6)(7,8)(10,14)(15,17)(16,22)(23,24)(25,27)(26,32)(28,31)(29,37)(30,36)(33,35)(34,40)(38,39);(1,3,5,4)(2,7,6,8)(9,11,13,19)(10,15,14,17)(12,20,21,18)(16,24,22,23)(25,28,31,27)(26,33,35,32)(29,37,30,36)(34,40,38,39);(1,3,5,4)(2,7,6,8)(9,12,13,21)(10,16,14,22)(11,18,19,20)(15,23,17,24)(25,29,31,30)(26,34,35,38)(27,36,28,37)(32,39,33,40);(10,14)(15,17)(16,22)(23,24)(25,30,31,29)(26,34,35,38)(27,36,28,37)(32,40,33,39);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

2^9|c^2=g,d^2=g,e^2=i,f^2=gh,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, 7614878);