#385201

order	512
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

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

979 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)(11,21)(12,26)(13,32)(14,20)(15,19)(16,24)(17,30)(18,22)(23,35)(25,34)(27,40)(28,38)(29,36)(31,37)(33,39);(9,11,22,25,15,13,14,24)(10,16,20,32,19,34,18,21)(12,27,31,23,30,39,29,28)(17,35,37,40,26,38,36,33);(1,2,4,3)(9,12)(10,17)(11,23)(13,28)(14,29)(15,30)(16,33)(18,36)(19,26)(20,37)(21,38)(22,31)(24,27)(25,39)(32,35)(34,40);(1,3,4,2)(9,13,15,11)(10,16,19,34)(12,28,30,23)(14,24,22,25)(17,33,26,40)(18,32,20,21)(27,31,39,29)(35,37,38,36)

Polycyclic presentation:

2^9|a^2=h,b^2=ei,c^2=fi,d^2=f,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, 292397);