#7708047

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

4501 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=gi,b^2=gi,c^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, 7615243);