#7304599

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

1326 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)(25,26,32,38)(27,33,42,50)(28,43,29,35)(30,37,31,36)(34,45,52,44)(39,53,40,48)(41,54,46,49)(47,55,51,56);(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)(29,39)(30,46)(31,41)(32,42)(34,51)(35,53)(36,54)(37,49)(38,50)(43,48)(44,55)(45,56)(47,52);(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)(27,39)(29,32)(30,44)(31,45)(33,47)(35,37)(36,43)(38,52)(40,42)(41,55)(46,56)(48,49)(50,51)(53,54);(2,6)(3,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(25,29,32,28)(26,35,38,43)(27,40,42,39)(30,44,31,45)(33,48,50,53)(34,36,52,37)(41,55,46,56)(47,54,51,49);(2,6)(5,7)(10,18)(15,19)(16,17)(23,24)(25,30,32,31)(26,36,38,37)(27,41,42,46)(28,44,29,45)(33,49,50,54)(34,43,52,35)(39,55,40,56)(47,48,51,53)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,d^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,e^a=efi,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

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7211795);