#6453761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437418
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	C₂⁵	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3107 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=i,e^2=gi,f^2=i,b^a=bf,c^a=cg,f^a=fi,c^b=ch,d^b=dh,f^b=fi,e^c=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, 6360957);