#420692

order	512
nilpotency class	3
isoclinism	isoclinic class of #417603
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

1171 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)(15,19)(16,20)(17,22)(18,24)(21,27)(23,33)(25,34)(26,35)(28,36)(29,37)(30,38)(31,40)(32,39)(41,44)(42,43);(5,7,9,12)(6,10,8,11)(13,15)(14,19)(16,29)(17,27)(18,28)(20,37)(21,22)(23,35)(24,36)(25,30)(26,33)(31,41)(32,42)(34,38)(39,43)(40,44);(2,4)(5,6,9,8)(7,10,12,11)(13,16,18,32)(14,20,24,39)(15,25,28,41)(17,31,33,30)(19,34,36,44)(21,37,35,43)(22,40,23,38)(26,42,27,29);(5,8,9,6)(7,10,12,11)(13,15)(14,21)(16,25)(17,26)(18,28)(19,22)(20,37)(23,36)(24,35)(27,33)(29,30)(31,42)(32,41)(34,40)(38,44)(39,43)

Polycyclic presentation:

2^9|a^2=ef,b^2=e,c^2=ei,d^2=e,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,g^b=gi,g^c=gi,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, 327888);