#424808

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1273 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)(5,6,8,10)(7,11,12,9)(13,14)(15,20)(16,21)(17,19)(18,29)(22,28)(23,34)(24,35)(25,32)(26,33)(27,38)(30,36)(31,37)(39,43)(40,44)(41,42);(14,19)(15,23,25,30)(20,34,32,36)(21,33)(22,29)(24,40,41,39)(35,44,42,43)(37,38);(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,24)(17,25)(18,30)(19,32)(21,35)(22,34)(23,28)(26,41)(27,39)(29,36)(31,40)(33,42)(37,44)(38,43);(2,4)(13,16,17,26)(14,21,19,33)(15,24,25,41)(18,31,28,27)(20,35,32,42)(22,38,29,37)(23,39,30,40)(34,43,36,44)

Polycyclic presentation:

2^9|a^2=f,b^2=ei,c^2=f,d^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,g^c=gi,g^d=gi

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, 332004);