#379985

order	512
nilpotency class	3
isoclinism	isoclinic class of #29748
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

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

1492 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)(13,14,19,23,17,22,18,33)(15,24,27,36,26,20,34,37)(16,28,32,40,30,21,31,39)(25,41,29,38,42,35,43,44);(6,9)(10,12)(13,15)(14,20)(16,29)(17,26)(18,34)(19,27)(21,38)(22,24)(23,37)(25,31)(28,44)(30,43)(32,42)(33,36)(35,39)(40,41);(2,4)(7,11)(10,12)(13,16)(14,21)(15,25)(17,30)(18,32)(19,31)(20,35)(22,28)(23,40)(24,41)(26,42)(27,43)(29,34)(33,39)(36,44)(37,38);(5,7,8,11)(6,10,9,12)(16,30)(21,28)(25,42)(29,43)(31,32)(35,41)(38,44)(39,40)

Polycyclic presentation:

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

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, 287181);