#6798325

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	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2293 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1300 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)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,29,33)(27,35,37,31)(28,39,40,55)(30,43,42,34)(32,46,47,41)(36,50,51,54)(38,45,53,49)(44,52,48,56);(3,4)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(25,27,29,37)(26,31,33,35)(28,36,40,51)(30,38,42,53)(32,44,47,48)(34,45,43,49)(39,54,55,50)(41,52,46,56);(1,3,5,4)(2,6,7,8)(25,28,30,41,29,40,42,46)(26,32,34,39,33,47,43,55)(27,36,38,52,37,51,53,56)(31,44,45,54,35,48,49,50);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(26,33)(27,37)(32,47)(34,43)(36,51)(38,53)(39,55)(52,56);(1,4,5,3)(2,6,7,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(27,37)(31,35)(36,51)(38,53)(44,48)(45,49)(50,54)(52,56)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=g,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, 6705521);