#7331472

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1303 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(9,10,14,18)(11,19,21,15)(12,16,13,17)(20,24,22,23)(25,26)(27,35)(28,33)(29,31)(30,43)(32,38)(34,42)(36,50)(37,49)(39,45)(40,54)(41,56)(44,53)(46,52)(47,55)(48,51);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(26,31)(28,39)(32,44)(34,43)(35,49)(36,40)(41,51)(52,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,38,29,37,42,53)(26,32,34,35,31,44,43,49)(28,40,41,55,39,36,51,52)(33,47,48,54,45,46,56,50);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,39)(26,33,31,45)(27,36,37,40)(30,41,42,51)(32,46,44,47)(34,48,43,56)(35,50,49,54)(38,52,53,55);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(26,31)(27,37)(28,39)(34,43)(38,53)(41,51)(46,47)(50,54)

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fgi,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,d^c=dhi,e^c=ei,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, 7238668);