#346962

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
prime radical	#18300	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2146 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);(1,3,5,4)(2,6,7,8)(9,11)(10,15)(12,20)(13,26)(14,21)(16,28)(17,32)(18,29)(19,24)(22,34)(23,37)(25,27)(30,40)(31,39)(33,35)(36,38);(9,12,14,22)(10,16,18,30)(11,20,21,34)(13,23,24,35)(15,28,29,40)(17,31,25,36)(19,33,26,37)(27,38,32,39)

Polycyclic presentation:

2^9|c^2=h,d^2=i,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, 254158);