#460078

order	512
nilpotency class	3
isoclinism	isoclinic class of #453948
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1183 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6,8,9)(7,11,12,10)(13,14,17,22)(15,24,27,20)(16,29,31,38)(18,23,19,34)(21,35,30,32)(25,33,37,28)(26,40,42,41)(36,44,39,43);(1,2,4,3)(6,9)(10,11)(13,15,17,27)(14,20,22,24)(16,26,31,42)(18,33,19,28)(21,36,30,39)(23,25,34,37)(29,41,38,40)(32,44,35,43);(2,3)(5,7,8,12)(6,10,9,11)(13,14)(15,25)(16,30)(17,22)(18,34)(19,23)(20,33)(21,31)(24,28)(26,41)(27,37)(29,35)(32,38)(36,43)(39,44)(40,42);(1,3,4,2)(7,12)(10,11)(13,16)(14,21)(15,26)(17,31)(18,32)(19,35)(20,36)(22,30)(23,38)(24,39)(25,40)(27,42)(28,43)(29,34)(33,44)(37,41)

Polycyclic presentation:

2^9|a^2=f,b^2=efi,c^2=fi,d^2=ef,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,g^b=gi,d^c=de,g^c=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 367274);