#6465441

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438955
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	#35328	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#532 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2628 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=gh,d^2=hi,g^2=i,b^a=bf,c^a=cg,g^a=gi,c^b=ch,d^b=dh,e^b=ei,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, 6372637);