#364690

order	512
nilpotency class	3
isoclinism	isoclinic class of #23903
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×#386
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1544 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)(3,6)(4,5)(7,8)(9,10,13,18,14,17,24,31)(11,15,21,27,20,26,34,39)(12,16,23,30,22,29,35,37)(19,28,33,40,32,25,36,38);(2,5)(6,8)(9,11,13,21,14,20,24,34)(10,15,18,27,17,26,31,39)(12,19,23,33,22,32,35,36)(16,28,30,40,29,25,37,38);(1,3,4,7)(2,6,5,8)(9,12)(10,16)(11,19)(13,23)(14,22)(15,25)(17,29)(18,30)(20,32)(21,33)(24,35)(26,28)(27,38)(31,37)(34,36)(39,40);(2,5)(6,8)(12,22)(16,29)(19,32)(23,35)(25,28)(30,37)(33,36)(38,40)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72303810, #72313120, #72313122
maximal normal subgroup of	#72303810, #72313120, #72313122
maximal quotient of	#72316997, #72317038

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 271886);