#420270

order	512
nilpotency class	3
isoclinism	isoclinic class of #417692
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

1074 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327466);