#7705007

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×#538	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4414 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)(3,7)(4,6)(5,8)(9,10)(11,17)(12,20)(13,15)(14,18)(16,22)(19,23)(21,24)(25,26,29,32)(27,30,34,38)(28,31,36,40)(33,37,35,39);(1,3,5,4)(2,6,8,7)(9,11,14,13)(10,15,18,17)(12,19,22,21)(16,23,20,24)(25,27)(26,30)(28,35)(29,34)(31,39)(32,38)(33,36)(37,40);(9,12)(10,16)(11,19)(13,21)(14,22)(15,23)(17,24)(18,20)(25,28)(26,31)(27,33)(29,36)(30,37)(32,40)(34,35)(38,39);(1,3,5,4)(2,7,8,6)(9,11,14,13)(10,17,18,15)(12,21,22,19)(16,23,20,24)(27,34)(28,36)(30,38)(31,40);(1,3,5,4)(2,7,8,6)(9,13,14,11)(10,17,18,15)(12,21,22,19)(16,24,20,23)(28,36)(31,40)(33,35)(37,39);(1,4,5,3)(2,6,8,7)

Polycyclic presentation:

2^9|a^2=i,b^2=g,d^2=g,e^2=g,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, 7612203);