#7706713

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	#50875	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4496 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,19)(13,22)(14,17)(16,21)(18,24)(20,23)(25,26,29,32)(27,30,33,37)(28,31,35,39)(34,38,36,40);(2,5)(6,8)(9,11,14,13)(10,15,17,22)(12,18,21,20)(16,23,19,24)(27,33)(28,35)(30,37)(31,39);(9,12)(10,16)(11,18)(13,20)(14,21)(15,23)(17,19)(22,24)(25,27)(26,30)(28,36)(29,33)(31,40)(32,37)(34,35)(38,39);(9,11,14,13)(10,15,17,22)(12,20,21,18)(16,24,19,23)(25,28,29,35)(26,31,32,39)(27,34,33,36)(30,38,37,40);(9,13,14,11)(10,15,17,22)(12,20,21,18)(16,23,19,24)(27,33)(30,37)(34,36)(38,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=h,d^2=hi,e^2=h,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, 7613909);