#7637866

order	512
nilpotency class	2
isoclinism	isoclinic class of #37644
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×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4610 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7545062);