#7641809

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

4966 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)(9,10,13,11)(12,15,16,14)(17,18,23,19)(20,26,29,24)(21,27,22,25)(28,31,30,32);(2,5)(6,8)(9,11,13,10)(12,15,16,14)(17,19,23,18)(20,26,29,24)(21,25,22,27)(28,31,30,32);(10,11)(14,15)(17,20)(18,24)(19,26)(21,28)(22,30)(23,29)(25,31)(27,32);(9,12,13,16)(10,14,11,15)(17,21,23,22)(18,25,19,27)(20,28,29,30)(24,31,26,32);(1,3,4,7)(2,6,5,8)(20,29)(24,26)(28,30)(31,32);(17,22,23,21)(18,25,19,27)(20,30,29,28)(24,31,26,32)

Polycyclic presentation:

2^9|a^2=h,b^2=h,d^2=h,e^2=g,f^2=hi,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, 7549005);