#383899

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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	C₂²×2₋¹⁺⁴	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1543 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)(7,9)(8,10)(11,12)(13,14)(15,25)(16,29)(17,35)(18,24)(19,23)(20,27)(21,33)(22,36)(26,39)(28,38)(30,44)(31,42)(32,40)(34,41)(37,43);(5,7,8,11)(6,9,10,12)(13,15,18,27,19,17,36,28)(14,20,22,25,23,38,24,35)(16,30,32,31,33,43,34,26)(21,39,40,37,29,42,41,44);(6,10)(9,12)(13,16,19,33)(14,21,23,29)(15,26,17,31)(18,32,36,34)(20,37,38,44)(22,40,24,41)(25,42,35,39)(27,30,28,43);(6,10)(9,12)(13,17)(14,20)(15,19)(16,31)(18,27)(21,37)(22,35)(23,38)(24,25)(26,33)(28,36)(29,44)(30,32)(34,43)(39,40)(41,42)

Polycyclic presentation:

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

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, 291095);