#7307312

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

1330 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)(25,26,30,35)(27,31,38,47)(28,33,40,51)(29,42,44,34)(32,43,50,41)(36,52,37,46)(39,56,55,48)(45,54,49,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,21)(13,19)(14,20)(16,24)(17,18)(22,23)(25,27,30,38)(26,31,35,47)(28,37,40,36)(29,39,44,55)(32,49,50,45)(33,46,51,52)(34,48,42,56)(41,54,43,53);(1,3)(2,5)(4,8)(6,7)(25,28,30,40)(26,32,35,50)(27,36,38,37)(29,43,44,41)(31,45,47,49)(33,34,51,42)(39,54,55,53)(46,56,52,48);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,19,20,21)(15,23,18,24)(25,28)(26,33)(27,37)(29,41)(30,40)(31,46)(32,42)(34,50)(35,51)(36,38)(39,54)(43,44)(45,48)(47,52)(49,56)(53,55);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,19,20,21)(15,24,18,23)(27,38)(28,40)(31,47)(32,50)(33,51)(39,55)(41,43)(48,56)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=i,d^2=f,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=eh,d^c=dh,e^c=ei,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, 7214508);