#7634833

order	512
nilpotency class	2
isoclinism	isoclinic class of #36702
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

4805 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)(3,7)(4,8)(5,6)(9,10,12,13)(11,15,16,14)(17,18)(19,27)(20,26)(21,25)(22,23)(24,29)(28,31)(30,32);(2,6)(7,8)(9,10)(11,14)(12,13)(15,16)(18,23)(24,27)(25,26)(31,32);(10,13)(14,15)(17,19)(18,24)(20,30)(21,28)(22,29)(23,27)(25,32)(26,31);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20,22,21)(18,25,23,26)(19,28,29,30)(24,31,27,32);(1,3,5,4)(2,7,6,8)(17,21,22,20)(18,26,23,25)(19,28,29,30)(24,31,27,32);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

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

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, 7542029);