#6817081

order	512
nilpotency class	3
isoclinism	isoclinic class of #6806671
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#42679	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2397 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1994 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:

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

Polycyclic presentation:

2^9|c^2=fgh,d^2=fi,e^2=fh,h^2=i,b^a=bf,c^a=cg,d^a=df,e^a=ei,c^b=ch,e^b=efh,h^b=hi,d^c=di

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6724277);