#425247

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	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1090 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)(5,6,8,10)(7,11,12,9)(13,14)(15,25)(16,20)(17,22)(18,23)(19,28)(21,34)(24,35)(26,41)(27,38)(29,42)(30,37)(31,44)(32,39)(33,40)(36,43);(1,2)(3,4)(13,15)(14,19)(16,31)(17,28)(18,30)(20,38)(21,41)(22,25)(23,42)(24,37)(26,33)(27,32)(29,35)(34,43)(36,40)(39,44);(5,7,8,12)(6,9,10,11)(13,16)(14,20)(15,26)(17,32)(18,34)(19,36)(21,23)(22,39)(24,40)(25,41)(27,29)(28,43)(30,31)(33,35)(37,44)(38,42);(1,3,4,2)(13,16)(14,21)(15,27)(17,32)(18,33)(19,36)(20,23)(22,40)(24,39)(25,41)(26,29)(28,31)(30,43)(34,35)(37,38)(42,44)

Polycyclic presentation:

2^9|a^2=f,c^2=f,d^2=ei,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, 332443);