#494952

order	512
nilpotency class	3
isoclinism	isoclinic class of #3008
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₂×C₄	C₂³×C₈
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₈×D₄
prime radical	C₂×#135	C₂×C₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×#346 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

595 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,d^2=e,f^2=i,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=di,f^a=fi,d^b=di,f^c=fi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402148);