#408027

order	512
nilpotency class	3
isoclinism	isoclinic class of #395242
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

1252 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)(7,10)(8,9)(11,12)(13,14,18,24)(15,25,28,21)(16,29,32,43)(17,34,19,23)(20,35,37,27)(22,33,40,31)(26,41,30,36)(38,42,39,44);(2,4)(6,9)(10,12)(13,15,17,27,18,28,19,35)(14,20,23,25,24,37,34,21)(16,26,31,42,32,30,33,44)(22,36,29,39,40,41,43,38);(2,4)(13,15,18,28)(14,21,24,25)(16,30,32,26)(17,27,19,35)(20,34,37,23)(22,39,40,38)(29,41,43,36)(31,44,33,42);(5,7,8,11)(6,10,9,12)(13,16,17,31,18,32,19,33)(14,22,23,29,24,40,34,43)(15,26,27,42,28,30,35,44)(20,36,25,39,37,41,21,38)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=efi,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,g^b=gi,d^c=di,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, 315223);