#346982

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	#62853	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3173 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:

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

Polycyclic presentation:

2^9|a^2=h,c^2=i,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, 254178);