#436675

order	512
nilpotency class	3
isoclinism	isoclinic class of #418438
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1204 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)(13,14,16,21,19,24,18,23)(15,25,26,20,29,35,28,37)(17,31,30,22,34,38,33,40)(27,36,41,43,42,44,32,39);(1,3,4,2)(5,7,8,12)(6,9,10,11)(13,15)(14,20)(16,28)(17,32)(18,26)(19,29)(21,25)(22,39)(23,35)(24,37)(27,33)(30,42)(31,36)(34,41)(38,44)(40,43);(2,3)(6,10)(9,11)(13,16,19,18)(14,21,24,23)(15,26,29,28)(17,30,34,33)(20,35,37,25)(22,38,40,31)(27,41,42,32)(36,43,44,39);(2,3)(7,12)(9,11)(13,17,18,33,19,34,16,30)(14,22,23,31,24,40,21,38)(15,27,28,32,29,42,26,41)(20,36,25,39,37,44,35,43)

Polycyclic presentation:

2^9|a^2=eg,b^2=f,c^2=i,d^2=ef,e^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,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, 343871);