#7304525

order	512
nilpotency class	3
isoclinism	isoclinic class of #7300916
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	#1999	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1376 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)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,39)(28,35)(29,45)(30,38)(31,37)(32,36)(33,42)(34,44)(40,49)(41,50)(43,51)(46,53)(47,52)(48,56)(54,55);(1,2)(3,7)(4,6)(5,8)(9,11,14,21)(10,15,18,19)(12,20,13,22)(16,23,17,24)(25,27)(26,33)(28,40)(29,46)(30,41)(31,42)(32,43)(34,48)(35,53)(36,50)(37,39)(38,51)(44,55)(45,49)(47,56)(52,54);(1,3)(2,5)(4,8)(6,7)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,31,29)(26,34,37,52)(27,40,42,46)(30,44,32,47)(33,48,39,54)(35,38,45,36)(41,55,43,56)(49,50,53,51);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,31,28)(26,35,37,45)(27,40,42,46)(30,44,32,47)(33,49,39,53)(34,38,52,36)(41,56,43,55)(48,50,54,51);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,30,31,32)(26,36,37,38)(27,41,42,43)(28,44,29,47)(33,50,39,51)(34,35,52,45)(40,55,46,56)(48,53,54,49)

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7211721);