#7708032

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	#51941	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4544 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)(11,17)(12,16)(13,15)(14,18)(19,23)(20,22)(21,24)(25,26,30,34)(27,35,36,31)(28,32,29,33)(37,40,38,39);(2,5)(6,8)(25,27,30,36)(26,31,34,35)(28,38,29,37)(32,40,33,39);(9,11)(10,14)(12,19)(13,18)(15,17)(16,21)(20,24)(22,23)(27,36)(28,29)(31,35)(32,33);(1,3,4,7)(2,6,5,8)(11,18)(14,17)(19,24)(21,23)(25,28,30,29)(26,32,34,33)(27,37,36,38)(31,39,35,40);(10,15)(14,17)(16,22)(21,23)(25,29,30,28)(26,33,34,32)(27,37,36,38)(31,39,35,40);(9,12,13,20)(10,16,15,22)(11,19,18,24)(14,21,17,23)

Polycyclic presentation:

2^9|a^2=i,b^2=i,d^2=g,e^2=i,f^2=h,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, 7615228);