#430340

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

1181 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)(16,28)(17,19)(18,23)(21,29)(22,31)(24,35)(25,39)(26,32)(27,36)(30,33)(34,40)(37,42)(38,44)(41,43);(5,7,8,12)(6,9,10,11)(14,19)(15,24)(16,29)(17,31)(22,23)(26,27)(28,33)(30,42)(32,35)(34,39)(38,40)(43,44);(6,10)(9,11)(13,15,18,27)(14,20,23,36)(16,25,30,41)(17,26,31,24)(19,32,22,35)(21,34,37,44)(28,39,33,43)(29,40,42,38);(2,4)(7,12)(9,11)(13,16,17,29,18,30,31,42)(14,21,22,28,23,37,19,33)(15,25,26,40,27,41,24,38)(20,34,35,39,36,44,32,43)

Polycyclic presentation:

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

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, 337536);