#7307378

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300980
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1326 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)(11,18)(12,16)(13,22)(14,17)(15,20)(19,24)(21,23)(25,26,30,36)(27,38,41,32)(28,34,45,54)(29,37,31,35)(33,46,52,44)(39,53,43,48)(40,49,42,50)(47,55,51,56);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,30,41)(26,32,36,38)(28,43,45,39)(29,40,31,42)(33,51,52,47)(34,53,54,48)(35,49,37,50)(44,56,46,55);(1,3)(2,5)(4,8)(6,7)(25,28,30,45)(26,33,36,52)(27,39,41,43)(29,44,31,46)(32,47,38,51)(34,37,54,35)(40,55,42,56)(48,50,53,49);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,34)(27,39)(29,46)(30,45)(31,44)(32,48)(33,35)(36,54)(37,52)(38,53)(40,56)(41,43)(42,55)(47,49)(50,51);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(25,29,30,31)(26,35,36,37)(27,40,41,42)(28,44,45,46)(32,49,38,50)(33,34,52,54)(39,55,43,56)(47,48,51,53)

Polycyclic presentation:

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

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7214574);