#353588

order	512
nilpotency class	3
isoclinism	isoclinic class of #17367
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂²×C₂²⋊C₄
commutator	C₂³	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₂×C₄×C₂²⋊C₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×#97 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1434 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)(3,6)(4,5)(7,8)(9,10)(11,15)(12,22,23,16)(13,18)(14,17)(19,31,32,26)(20,27)(21,28)(24,35,36,29)(25,30)(33,39,40,37)(34,38);(2,5)(6,8)(9,11,13,21,14,20,25,34)(10,15,18,28,17,27,30,38)(12,19,24,33,23,32,36,40)(16,26,29,37,22,31,35,39);(9,12)(10,16)(11,19)(13,24)(14,23)(15,26)(17,22)(18,29)(20,32)(21,33)(25,36)(27,31)(28,37)(30,35)(34,40)(38,39);(1,3,4,7)(2,6,5,8)(11,20)(15,27)(19,32)(21,34)(26,31)(28,38)(33,40)(37,39)

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72224727
maximal normal subgroup of	#72224727

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260784);