#6438538

order	512
nilpotency class	3
isoclinism	isoclinic class of #6437521
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35150	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2291 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6345734);