#7201331

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200022
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	#53065	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1459 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fi,c^2=fi,e^2=i,g^2=i,b^a=bf,c^a=cg,d^a=dhi,e^a=ef,g^a=gi,c^b=ch,d^b=dfh,e^b=eh,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, 7108527);