#7331216

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

1301 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)(30,42)(32,38)(34,41)(36,48)(37,47)(39,44)(40,53)(43,51)(45,50)(46,52)(49,55)(54,56);(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)(32,43)(34,42)(35,47)(36,49)(40,52)(50,56);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,27,30,38,29,37,41,51)(26,32,34,35,31,43,42,47)(28,36,40,50,39,49,52,56)(33,45,46,48,44,54,53,55);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28,29,39)(26,33,31,44)(27,36,37,49)(30,40,41,52)(32,45,43,54)(34,46,42,53)(35,48,47,55)(38,50,51,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(26,31)(27,37)(28,39)(34,42)(38,51)(40,52)(45,54)(48,55)

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=dh,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, 7238412);