#369732

order	512
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂²×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	D₄²
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	H(5,𝔽₂)
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1066 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)(7,12)(8,10)(9,11)(13,14)(15,25)(16,30)(17,21)(18,23)(19,24)(20,28)(22,32)(26,36)(27,41)(29,35)(31,37)(33,38)(34,43)(39,42)(40,44);(1,3,4,2)(5,7,9,8)(6,10,11,12)(13,15,17,28,19,29,18,26)(14,20,23,25,24,36,21,35)(16,27,32,42,31,43,33,40)(22,34,30,39,38,41,37,44);(5,8,9,7)(6,10,11,12)(14,21,24,23)(15,26,29,28)(16,31)(22,30,38,37)(27,42,43,40)(32,33)(34,41)(39,44);(6,11)(10,12)(13,16,17,32,19,31,18,33)(14,22,23,30,24,38,21,37)(15,27,28,42,29,43,26,40)(20,34,25,39,36,41,35,44)

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=eg,d^2=f,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,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, 276928);