#7702877

order	512
nilpotency class	2
isoclinism	isoclinic class of #45438
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	#45770	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4493 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)(9,10,13,16)(11,14,18,22)(12,19,20,15)(17,23,24,21)(25,26,29,32)(27,33,34,30)(28,36,37,31)(35,38,40,39);(2,5)(6,8)(9,11,13,18)(10,14,16,22)(12,17,20,24)(15,21,19,23)(27,34)(30,33)(35,40)(38,39);(9,12)(10,15)(11,17)(13,20)(14,21)(16,19)(18,24)(22,23)(25,27)(26,30)(28,35)(29,34)(31,38)(32,33)(36,39)(37,40);(12,20)(15,19)(17,24)(21,23)(25,28,29,37)(26,31,32,36)(27,35,34,40)(30,38,33,39);(10,16)(14,22)(15,19)(21,23)(26,32)(27,34)(31,36)(35,40);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=h,b^2=hi,d^2=i,f^2=g,b^a=bg,c^a=ch,d^a=di,e^a=eh,c^b=ci,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, 7610073);