#430316

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

1208 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=ef,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,h^b=hi,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, 337512);