#6456614

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437547
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	#2431	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1889 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=i,c^2=f,d^2=f,e^2=gi,h^2=i,b^a=bf,c^a=cg,c^b=ch,d^b=dhi,h^b=hi,e^c=ei,h^c=hi,h^d=hi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6363810);