#6466221

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438975
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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2296 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2198 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)(5,6)(7,11)(8,9)(10,12)(13,14,18,22,19,23,32,38)(15,24,16,27,26,37,29,20)(17,21,31,36,30,35,39,43)(25,34,28,41,40,44,42,33);(2,4)(6,9)(10,11)(13,15,19,26)(14,20,23,27)(16,18,29,32)(17,28,30,42)(21,34,35,44)(22,37,38,24)(25,39,40,31)(33,43,41,36);(5,7,8,12)(6,10,9,11)(13,16,18,26,19,29,32,15)(14,20,22,24,23,27,38,37)(17,28,31,40,30,42,39,25)(21,33,36,34,35,41,43,44);(13,17,18,31,19,30,32,39)(14,21,22,36,23,35,38,43)(15,25,16,28,26,40,29,42)(20,33,24,34,27,41,37,44);(17,30)(21,35)(25,40)(28,42)(31,39)(33,41)(34,44)(36,43)

Polycyclic presentation:

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

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, 6373417);