#494506

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	C₂³×D₁₆	C₂
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1480 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)(9,10,12,18,26,33,39,31,14,17,27,32,13,19,25,16)(11,20,22,35,28,40,38,30,21,36,37,29,23,34,24,15);(1,2,3,4)(9,11)(10,15)(12,24)(13,28)(14,21)(16,20)(17,30)(18,34)(19,35)(22,25)(23,26)(27,38)(29,33)(31,36)(32,40)(37,39);(6,8)(10,16)(12,25)(13,26)(15,20)(17,31)(18,19)(22,24)(23,28)(27,39)(29,40)(30,36)(32,33)(34,35)(37,38);(10,17)(11,21)(16,31)(18,32)(19,33)(22,37)(23,28)(24,38)

Polycyclic presentation:

2^9|a^2=g,b^2=egh,e^2=hi,f^2=hi,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=di,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, #72320769, #72320837, …
maximal normal subgroup of	#72320035, #72320769, #72320837, …
maximal quotient of	#72322822

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 401702);