#7628635

order	512
nilpotency class	2
isoclinism	isoclinic class of #34980
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×#535	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5282 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7535831);