#7639568

order	512
nilpotency class	2
isoclinism	isoclinic class of #40484
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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4955 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)(17,18,23,26)(19,24,20,28)(21,30,22,25)(27,32,29,31);(2,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,26,28)(21,29,22,27)(25,32,30,31);(9,11)(10,13)(12,16)(14,15)(17,20,23,19)(18,24,26,28)(21,29,22,27)(25,31,30,32);(10,14)(13,15)(17,21,23,22)(18,25,26,30)(19,27,20,29)(24,31,28,32);(11,16)(13,15)(17,22,23,21)(18,25,26,30)(19,27,20,29)(24,32,28,31);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

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

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, 7546764);