#364367

order	512
nilpotency class	3
isoclinism	isoclinic class of #189674
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	C₂⁵	#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ #2280 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1038 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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)(2,5)(4,8)(6,7)(9,11,15,13,16,24,14,23)(10,17,32,38,21,31,20,19)(12,26,29,39,30,22,28,27)(18,34,37,35,36,33,25,40);(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=eg,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,d^b=di,h^c=hi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226664, #72226674
maximal normal subgroup of	#72226664, #72226674

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271563);