#390148

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
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

969 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)(6,9)(7,11)(13,14)(15,20)(16,29)(17,31)(18,24)(19,25)(21,28)(22,33)(23,32)(26,40)(27,36)(30,35)(34,38)(37,41)(39,44)(42,43);(5,6,8,9)(7,12,11,10)(13,15)(14,20)(16,18)(17,26)(19,28)(21,25)(22,35)(23,36)(24,29)(27,32)(30,33)(31,40)(34,42)(37,39)(38,43)(41,44);(1,3,4,2)(13,16,18,28,19,27,32,15)(14,21,23,29,25,20,24,36)(17,30,33,42,34,41,44,26)(22,37,31,43,39,35,38,40);(5,7,8,11)(6,10,9,12)(13,17,32,44,19,34,18,33)(14,22,24,38,25,39,23,31)(15,26,27,41,28,42,16,30)(20,35,29,43,21,37,36,40)

Polycyclic presentation:

2^9|a^2=h,b^2=e,c^2=fh,d^2=efi,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=def,f^a=fi,c^b=cg,d^b=de,g^b=gi,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 297344);