#375772

order	512
nilpotency class	3
isoclinism	isoclinic class of #23886
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

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

1651 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,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,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,e^d=ei,f^d=fi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 282968);