#7037554

order	512
nilpotency class	3
isoclinism	isoclinic class of #6994139
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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2395 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1244 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=i,c^2=i,d^2=i,e^2=fi,g^2=i,b^a=bf,c^a=cg,e^a=eh,g^a=gi,c^b=ch,d^b=df,e^b=eh,d^c=dhi,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6944750);