#7641784

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

4958 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)(3,7)(4,6)(5,8)(9,10)(11,15)(12,13)(14,16)(17,18)(19,23)(20,27)(21,25)(22,26)(24,29)(28,32)(30,31);(1,3,5,4)(2,6,8,7)(9,10)(11,14)(12,13)(15,16)(17,19,23,18)(20,27,29,24)(21,26,22,25)(28,32,30,31);(10,13)(14,15)(17,20)(18,24)(19,27)(21,30)(22,28)(23,29)(25,32)(26,31);(1,3,5,4)(2,7,8,6)(9,11,12,16)(10,14,13,15)(18,19)(24,27)(25,26)(31,32);(1,4,5,3)(2,6,8,7)(17,21,23,22)(18,25,19,26)(20,28,29,30)(24,31,27,32);(17,22,23,21)(18,26,19,25)(20,28,29,30)(24,31,27,32)

Polycyclic presentation:

2^9|b^2=g,d^2=gh,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, 7548980);