#6460064

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	C₂×#535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2297 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2300 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:

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

Polycyclic presentation:

2^9|a^2=fi,c^2=ghi,d^2=f,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, 6367260);