#6370870

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂¹⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
socle	C₂²	C₂³×C₂²⋊C₄
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×#41 ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

4446 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=h,c^2=f,e^2=i,b^a=bf,c^a=cg,g^a=gi,d^b=di,h^c=hi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313694
maximal normal subgroup of	#72313694
maximal quotient of	#72314042

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6278066);