#457311

order	512
nilpotency class	3
isoclinism	isoclinic class of #454426
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×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

947 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)(7,10)(8,9)(11,12)(13,14,20,37,18,24,19,25)(15,26,30,40,28,21,29,44)(16,31,36,22,27,42,33,39)(17,34,38,43,35,41,32,23);(1,2)(3,4)(6,9)(10,12)(13,15,20,30,18,28,19,29)(14,21,37,44,24,26,25,40)(16,32,36,17,27,38,33,35)(22,41,42,23,39,34,31,43);(2,4)(6,9)(7,11)(13,16)(14,22)(15,17)(18,27)(19,36)(20,33)(21,23)(24,39)(25,42)(26,43)(28,35)(29,38)(30,32)(31,37)(34,40)(41,44);(5,7,8,11)(6,10,9,12)(13,17)(14,23)(15,27)(16,28)(18,35)(19,32)(20,38)(21,39)(22,26)(24,43)(25,41)(29,36)(30,33)(31,44)(34,37)(40,42)

Polycyclic presentation:

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

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, 364507);