#6816058

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682360
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	#37659	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2174 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=i,e^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6723254);