#7228804

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	C₂×#547	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2398 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1398 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=fi,d^2=fhi,e^2=fi,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, 7136000);