#7211700

order	512
nilpotency class	3
isoclinism	isoclinic class of #7207866
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#52977	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1354 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=fi,c^2=fg,d^2=fi,e^2=f,g^2=i,b^a=bf,c^a=cg,d^a=dh,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7118896);