#386797

order	512
nilpotency class	3
isoclinism	isoclinic class of #30490
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

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

1156 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 293993);