#7337793

order	512
nilpotency class	3
isoclinism	isoclinic class of #7324436
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	#25534	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1389 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)(27,35)(28,39)(29,33)(30,34)(31,38)(32,36)(37,47)(40,52)(41,49)(42,44)(43,48)(45,50)(46,51)(53,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11)(10,15)(12,19)(13,20)(14,21)(16,23)(17,18)(22,24)(25,27,30,38)(26,31,34,35)(28,36,41,51)(29,42,43,37)(32,39,46,49)(33,47,48,44)(40,53,54,50)(45,55,56,52);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,17,22)(11,19,21,20)(15,23,18,24)(25,28,27,36,30,41,38,51)(26,32,31,39,34,46,35,49)(29,40,37,50,43,54,42,53)(33,45,44,52,48,56,47,55);(2,6)(3,8)(9,12,14,13)(10,16,17,22)(11,20,21,19)(15,24,18,23)(25,29,30,43)(26,33,34,48)(27,37,38,42)(28,40,41,54)(31,44,35,47)(32,45,46,56)(36,50,51,53)(39,52,49,55);(2,6)(5,7)(9,13,14,12)(10,16,17,22)(11,20,21,19)(15,23,18,24)(26,34)(29,43)(31,35)(32,46)(37,42)(39,49)(40,54)(50,53)

Polycyclic presentation:

2^9|b^2=i,c^2=fgi,d^2=f,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=eh,d^c=dh,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7244989);