#7238242

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

1297 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)(25,26,30,36)(27,38,42,32)(28,34,44,52)(29,37,31,35)(33,46,50,43)(39,48,40,51)(41,49,45,53)(47,56,54,55);(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)(26,32)(28,39)(29,45)(30,42)(31,41)(33,47)(34,51)(35,53)(36,38)(37,49)(40,44)(43,56)(46,55)(48,52)(50,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,30,44)(26,33,36,50)(27,39,42,40)(29,43,31,46)(32,47,38,54)(34,37,52,35)(41,55,45,56)(48,53,51,49);(3,8)(5,7)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,28)(26,34)(27,40)(29,46)(30,44)(31,43)(32,48)(33,35)(36,52)(37,50)(38,51)(39,42)(41,55)(45,56)(47,49)(53,54);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,29,30,31)(26,35,36,37)(27,41,42,45)(28,43,44,46)(32,49,38,53)(33,34,50,52)(39,55,40,56)(47,48,54,51)

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=f,d^2=fi,e^2=i,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, 7145438);