#7242428

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

1302 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=i,d^2=fi,e^2=fi,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, 7149624);