#494447

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	C₂×#3058	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×#41 ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1504 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)(9,10,12,17,24,32,36,28,14,19,25,27,13,18,23,15)(11,16,21,30,34,39,40,37,20,29,33,38,22,31,35,26);(1,2,3,4)(10,15)(12,23)(13,24)(16,26)(17,18)(19,28)(21,35)(22,34)(25,36)(27,32)(29,37)(30,31)(33,40)(38,39);(6,8)(9,11)(10,16)(12,21)(13,22)(14,20)(15,26)(17,30)(18,31)(19,29)(23,35)(24,34)(25,33)(27,38)(28,37)(32,39)(36,40);(11,20)(16,29)(21,33)(22,34)(26,37)(30,38)(31,39)(35,40)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321432, #72321861, #72321869
maximal normal subgroup of	#72321432, #72321861, #72321869
maximal quotient of	#72322840

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401643);