#444349

order	512
nilpotency class	3
isoclinism	isoclinic class of #418500
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1198 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=f,c^2=gi,d^2=ef,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,g^b=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, 351545);