#6516884

order	512
nilpotency class	3
isoclinism	isoclinic class of #6499792
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×#3547	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3304 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)(15,24)(16,28)(17,22)(18,32)(19,23)(20,26)(21,30)(25,40)(27,35)(29,38)(31,37)(33,36)(34,41)(39,43)(42,44);(5,6)(7,10)(8,9)(11,12)(13,15,19,27)(14,20,23,36)(16,25,30,43)(17,26,18,33)(21,34,28,44)(22,35,32,24)(29,41,31,42)(37,39,38,40);(2,4)(6,9)(7,11)(13,16)(14,21)(15,25)(17,31)(18,29)(19,30)(20,34)(22,38)(23,28)(24,39)(26,42)(27,43)(32,37)(33,41)(35,40)(36,44);(5,7,8,11)(6,10,9,12);(6,9)(10,12)(13,17,19,18)(14,22,23,32)(15,26,27,33)(16,29,30,31)(20,35,36,24)(21,37,28,38)(25,41,43,42)(34,39,44,40)

Polycyclic presentation:

2^9|a^2=gi,b^2=i,d^2=h,e^2=i,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,e^b=eh,f^b=fi,d^c=dh,e^c=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, 6424080);