#413453

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

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

984 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:

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

Polycyclic presentation:

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

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, 320649);