#7201183

order	512
nilpotency class	3
isoclinism	isoclinic class of #7200042
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	#2388	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2388 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1418 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)(11,17)(12,15)(13,16)(14,19)(18,23)(20,22)(21,24)(25,26,31,36)(27,32,41,50)(28,42,44,34)(29,35,30,46)(33,45,51,43)(37,48,38,52)(39,49,40,55)(47,54,56,53);(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)(28,37)(29,39)(30,40)(31,41)(33,47)(34,52)(35,49)(36,50)(38,44)(42,48)(43,53)(45,54)(46,55)(51,56);(3,7)(5,8)(9,12,13,20)(10,15,16,22)(11,18,19,24)(14,21,17,23)(25,28)(26,33)(27,37)(29,43)(30,45)(31,44)(32,47)(34,35)(36,51)(38,41)(39,53)(40,54)(42,46)(48,55)(49,52)(50,56);(2,6)(3,7)(11,19)(14,17)(18,24)(21,23)(25,28)(26,34)(27,38)(29,45)(30,43)(31,44)(32,48)(33,35)(36,42)(37,41)(39,53)(40,54)(46,51)(47,55)(49,56)(50,52);(3,7)(5,8)(10,16)(14,17)(15,22)(21,23)(25,29,31,30)(26,35,36,46)(27,39,41,40)(28,43,44,45)(32,49,50,55)(33,34,51,42)(37,53,38,54)(47,52,56,48)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=f,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=dfhi,e^b=eh,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, 7108379);