#6442766

order	512
nilpotency class	3
isoclinism	isoclinic class of #6438949
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#35149	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#525 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

4596 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315555, #72315560, #72315572
maximal normal subgroup of	#72315555, #72315560, #72315572
maximal quotient of	#72315686

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6349962);