#6798429

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	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2535	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1358 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)(3,8)(4,6)(5,7)(9,10)(11,19)(12,16)(13,17)(14,18)(15,21)(20,24)(22,23)(25,26)(27,35)(28,39)(29,33)(30,43)(31,37)(32,42)(34,40)(36,49)(38,45)(41,54)(44,56)(46,55)(47,52)(48,50)(51,53);(3,4)(6,8)(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)(39,53)(40,50)(41,51)(42,52)(43,48)(46,47)(49,54)(55,56);(1,3,5,4)(2,6,7,8)(9,12,14,13)(10,16,18,17)(11,20,21,22)(15,23,19,24)(25,28,40,55,29,41,30,42)(26,32,43,54,33,46,34,39)(27,36,50,56,37,51,38,52)(31,44,48,49,35,47,45,53);(9,12,14,13)(10,17,18,16)(11,20,21,22)(15,24,19,23)(26,33)(27,37)(32,46)(34,43)(36,51)(38,50)(39,54)(52,56);(1,4,5,3)(2,6,7,8)(9,13,14,12)(10,17,18,16)(11,20,21,22)(15,23,19,24)(27,37)(31,35)(36,51)(38,50)(44,47)(45,48)(49,53)(52,56)

Polycyclic presentation:

2^9|c^2=fg,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, 6705625);