#407981

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1294 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)(13,14,18,25,20,24,19,37)(15,21,28,40,27,42,30,41)(16,29,32,39,31,43,36,22)(17,33,34,44,35,38,26,23);(1,2)(3,4)(6,9)(10,12)(13,15,19,28,20,27,18,30)(14,21,25,41,24,42,37,40)(16,26,32,35,31,34,36,17)(22,38,43,23,39,33,29,44);(2,4)(13,16)(14,22)(15,26)(17,30)(18,36)(19,32)(20,31)(21,38)(23,41)(24,39)(25,43)(27,34)(28,35)(29,37)(33,42)(40,44);(5,7,8,11)(6,10,9,12)(13,17,18,34,20,35,19,26)(14,23,37,38,24,44,25,33)(15,16,30,36,27,31,28,32)(21,39,40,43,42,22,41,29)

Polycyclic presentation:

2^9|a^2=f,b^2=h,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 315177);