#6810621

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806623
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#392	D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2392 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1768 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 44 points:

(1,2)(3,4)(6,10)(7,8);(5,6)(7,12)(8,11)(9,10)(13,14,18,22)(15,23,25,37)(16,28,30,42)(17,31,24,21)(19,32,33,27)(20,35,36,29)(26,40,39,34)(38,43,44,41);(2,4)(5,7,9,8)(6,11,10,12)(13,15,24,32,18,25,17,27)(14,19,31,37,22,33,21,23)(16,26,35,43,30,39,29,41)(20,34,42,44,36,40,28,38);(5,8,9,7)(6,11,10,12)(15,25)(19,33)(23,37)(26,39)(27,32)(34,40)(38,44)(41,43);(13,16,24,35,18,30,17,29)(14,20,31,42,22,36,21,28)(15,26,32,43,25,39,27,41)(19,34,37,44,33,40,23,38)

Polycyclic presentation:

2^9|b^2=i,c^2=hi,d^2=f,e^2=fhi,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6717817);