#441104

order	512
nilpotency class	3
isoclinism	isoclinic class of #417464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1496 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)(6,10)(11,12)(13,14,19,22,17,24,18,23)(15,20,28,36,26,37,27,35)(16,29,33,39,31,38,32,21)(25,40,42,44,30,43,41,34);(5,6,9,10)(7,11,8,12)(13,15)(14,20)(16,30)(17,26)(18,28)(19,27)(21,34)(22,35)(23,36)(24,37)(25,31)(29,43)(32,41)(33,42)(38,40)(39,44);(2,4)(5,7,9,8)(6,11,10,12)(13,16)(14,21,24,39)(15,25)(17,31)(18,32)(19,33)(20,34,37,44)(22,29,23,38)(26,30)(27,41)(28,42)(35,43,36,40);(5,8,9,7)(6,11,10,12)(14,22,24,23)(15,26)(16,31)(20,35,37,36)(21,38,39,29)(27,28)(32,33)(34,40,44,43)

Polycyclic presentation:

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

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, 348300);