#6512887

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499823
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#3554	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#529 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3125 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)(13,14,20,26)(15,27,17,23)(16,31,35,40)(18,25,19,24)(21,30,37,29)(22,33,41,34)(28,38,32,36)(39,43,42,44);(2,4)(5,6)(7,10)(8,9)(11,12)(13,15,20,17)(14,21,26,37)(16,32,35,28)(18,30,19,29)(22,38,41,36)(23,25,27,24)(31,39,40,42)(33,43,34,44);(6,9)(7,11)(13,16,20,35)(14,22,26,41)(15,28,17,32)(18,34,19,33)(21,36,37,38)(23,42,27,39)(24,31,25,40)(29,44,30,43);(5,7,8,11)(6,10,9,12)(13,17,20,15)(14,23,26,27)(16,28,35,32)(18,30,19,29)(21,25,37,24)(22,39,41,42)(31,38,40,36)(33,43,34,44);(6,9)(10,12)(15,17)(21,37)(23,27)(28,32)(29,30)(36,38)(39,42)(43,44)

Polycyclic presentation:

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

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, 6420083);