#6658600

order	512
nilpotency class	3
isoclinism	isoclinic class of #6632048
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂×#121	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2298 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1330 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6565796);