#7641525

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

4893 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)(3,6)(4,5)(7,8)(11,12)(14,16)(17,18,22,25)(19,23,27,32)(20,28,21,24)(26,31,29,30);(2,5)(6,8)(9,10,13,15)(11,16,12,14)(17,19,22,27)(18,23,25,32)(20,29,21,26)(24,31,28,30);(9,11,13,12)(10,14,15,16)(17,18)(19,23)(20,24)(21,28)(22,25)(26,30)(27,32)(29,31);(9,12,13,11)(10,14,15,16)(17,20,22,21)(18,24,25,28)(19,26,27,29)(23,30,32,31);(17,21,22,20)(18,24,25,28)(19,26,27,29)(23,31,32,30);(1,3,4,7)(2,6,5,8)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=i,d^2=h,e^2=hi,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, 7548721);