#369283

order	512
nilpotency class	3
isoclinism	isoclinic class of #24468
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1352 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:

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

Polycyclic presentation:

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

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, 276479);