#346958

order	512
nilpotency class	3
isoclinism	isoclinic class of #346950
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2141 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 254154);