#7705757

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	#49562	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4985 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)(11,15)(12,14)(13,16)(17,18,21,24)(19,22,26,30)(20,23,28,32)(25,29,27,31);(2,5)(6,8)(17,19,21,26)(18,22,24,30)(20,27,28,25)(23,31,32,29);(9,11,12,16)(10,13,14,15)(17,20)(18,23)(19,25)(21,28)(22,29)(24,32)(26,27)(30,31);(11,16)(13,15)(19,26)(20,28)(22,30)(23,32);(10,14)(13,15)(20,28)(23,32)(25,27)(29,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=h,f^2=g,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, 7612953);