#7238243

order	512
nilpotency class	3
isoclinism	isoclinic class of #7234437
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)(25,26,30,35)(27,31,37,44)(28,33,39,48)(29,32,41,34)(36,43,49,55)(38,42,51,45)(40,46,52,47)(50,53,56,54);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,31)(28,36)(29,38)(30,37)(32,42)(33,43)(34,45)(35,44)(39,49)(40,50)(41,51)(46,53)(47,54)(48,55)(52,56);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(26,32)(28,39)(29,41)(31,42)(33,47)(34,35)(36,49)(38,51)(43,54)(44,45)(46,48)(53,55);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,33)(27,36)(29,40)(30,39)(31,43)(32,46)(34,47)(35,48)(37,49)(38,50)(41,52)(42,53)(44,55)(45,54)(51,56);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(27,37)(28,39)(31,44)(33,48)(38,51)(40,52)(42,45)(46,47)

Polycyclic presentation:

2^9|a^2=fi,b^2=f,c^2=f,d^2=f,g^2=i,b^a=bf,c^a=cg,e^a=ef,g^a=gi,c^b=ch,d^b=df,e^b=ehi,d^c=dhi,g^c=gi,e^d=ei

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7145439);