#7639626

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

4962 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^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, 7546822);