#6854618

order	512
nilpotency class	3
isoclinism	isoclinic class of #6682380
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2295 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1138 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=fgh,d^2=fi,e^2=fhi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efh

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6761814);