#494504

order	512
nilpotency class	4
isoclinism	isoclinic class of #33522
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂²×#509	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1410 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=i,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^b=di,e^b=eh,f^b=fh,g^b=gh,h^b=hi,e^c=eh,f^c=fh,g^c=gh,h^c=hi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72320035, #72320798, #72320810, …
maximal normal subgroup of	#72320035, #72320798, #72320810, …
maximal quotient of	#72322827

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401700);