#363546

order	512
nilpotency class	3
isoclinism	isoclinic class of #24278
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#121
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#380
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

996 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=e,d^2=eg,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,f^c=fi

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 270742);