#7337991

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324500
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	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1361 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)(15,21)(19,24)(20,23)(25,26,29,31)(27,35,36,51)(28,33,41,45)(30,43,37,34)(32,46,44,39)(38,50,40,54)(42,56,52,49)(47,55,48,53);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(26,31)(27,36)(33,45)(34,43)(38,40)(39,46)(49,56)(53,55);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,27,37,46,29,36,30,39)(26,32,43,51,31,44,34,35)(28,40,52,53,41,38,42,55)(33,48,56,50,45,47,49,54);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,28)(26,33)(27,38)(29,41)(30,42)(31,45)(32,47)(34,49)(35,50)(36,40)(37,52)(39,53)(43,56)(44,48)(46,55)(51,54);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,31)(27,36)(28,41)(34,43)(39,46)(42,52)(47,48)(50,54)

Polycyclic presentation:

2^9|a^2=i,c^2=fg,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=dfh,e^b=ehi,d^c=dh,e^c=ei,e^d=ei

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7245187);