#7639619

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

4840 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)(17,18)(19,21)(20,28)(22,25)(23,27)(24,30)(26,29)(31,32);(2,6)(7,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,23)(20,26)(21,25)(22,27)(24,29)(28,31)(30,32);(1,3,5,4)(2,7,6,8)(9,11)(10,13)(12,16)(14,15)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,31,23,29);(10,14)(13,15)(18,25)(19,27)(24,28)(26,32);(1,3,5,4)(2,7,6,8)(11,16)(13,15)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,32,28,26);(1,4,5,3)(2,8,6,7)

Polycyclic presentation:

2^9|c^2=g,e^2=g,f^2=gh,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, 7546815);