#7628364

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	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

5102 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=i,f^2=h,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, 7535560);