#401184

order	512
nilpotency class	3
isoclinism	isoclinic class of #394799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

988 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 308380);