#7640512

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

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

Polycyclic presentation:

2^9|a^2=h,c^2=h,d^2=hi,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, 7547708);