#6516823

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
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3352 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)(13,14,18,24,19,22,17,23)(15,25,29,20,27,37,28,36)(16,30,34,39,32,40,33,21)(26,38,43,44,31,35,42,41);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,22,36)(16,31,32,26)(17,29,18,28)(21,38,39,35)(23,37,24,25)(30,41,40,44)(33,42,34,43);(6,9)(7,11)(13,16)(14,21)(15,26)(17,34)(18,33)(19,32)(20,35)(22,39)(23,30)(24,40)(25,41)(27,31)(28,43)(29,42)(36,38)(37,44);(5,7,8,11)(6,10,9,12)(15,27)(16,32)(20,36)(21,39)(25,37)(28,29)(30,40)(33,34);(6,9)(10,12)(14,22)(15,27)(21,39)(23,24)(26,31)(28,29)(30,40)(42,43)

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=hi,f^2=i,g^2=i,b^a=bf,c^a=cg,e^a=ei,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, 6424019);