#7304759

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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2396 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1331 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=f,d^2=fi,e^2=i,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,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, 7211955);