#363839

order	512
nilpotency class	3
isoclinism	isoclinic class of #189517
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#382
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2285 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

764 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)(3,7)(4,6)(5,8)(9,10,15,20,16,21,14,32)(11,19,13,31,24,38,23,17)(12,25,28,18,30,37,29,36)(22,34,39,35,26,33,27,40);(1,3,4,8)(2,5,6,7)(9,11,14,23,16,24,15,13)(10,17,20,19,21,31,32,38)(12,26,28,27,30,22,29,39)(18,34,25,40,36,33,37,35);(2,6)(3,8)(9,12)(10,18,21,36)(11,22)(13,27)(14,28)(15,29)(16,30)(17,33,31,34)(19,35,38,40)(20,25,32,37)(23,39)(24,26);(2,6)(5,7)(9,13,15,24,16,23,14,11)(10,19,32,17,21,38,20,31)(12,27,29,26,30,39,28,22)(18,35,37,33,36,40,25,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226664, #72226669
maximal normal subgroup of	#72226664, #72226669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271035);