#6774652

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
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	C₂⁴	C₂²×D₄
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1187 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=fi,c^2=f,d^2=fh,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=di,e^b=efhi,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, 6681848);