#6698401

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667772
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 ◁ #2402 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1157 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=fi,d^2=g,e^2=i,h^2=i,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,h^b=hi,h^c=hi,e^d=ei,h^e=hi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6605597);