#7641488

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
prime radical	#42798	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4656 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:

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

Polycyclic presentation:

2^9|a^2=g,c^2=i,d^2=g,e^2=gi,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, 7548684);