#6516794

order	512
nilpotency class	3
isoclinism	isoclinic class of #6500076
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸×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×#3561	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3318 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:

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

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,d^a=di,c^b=ch,d^b=di,e^b=eh,f^b=fi,g^b=gi,d^c=dh,f^c=fi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6423990);