#6954141

order	512
nilpotency class	3
isoclinism	isoclinic class of #6884969
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

1373 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)(3,7)(4,6)(5,8)(10,14)(12,19)(15,21)(17,20)(25,26,31,36)(27,32,39,48)(28,41,29,34)(30,44,45,35)(33,43,51,42)(37,52,38,47)(40,56,55,49)(46,54,50,53);(1,3,4,8)(2,5,6,7)(9,10,13,14)(11,15,18,21)(12,16,19,22)(17,23,20,24)(25,27)(26,32)(28,38)(29,37)(30,40)(31,39)(33,50)(34,52)(35,49)(36,48)(41,47)(42,54)(43,53)(44,56)(45,55)(46,51);(3,8)(5,7)(9,11,13,18)(10,15,14,21)(12,20,19,17)(16,24,22,23)(25,28)(26,33)(27,37)(29,31)(30,43)(32,46)(34,35)(36,51)(38,39)(40,54)(41,44)(42,45)(47,49)(48,50)(52,56)(53,55);(2,6)(5,7)(9,12,13,19)(10,16,14,22)(11,17,18,20)(15,23,21,24)(25,29,31,28)(26,34,36,41)(27,38,39,37)(30,43,45,42)(32,47,48,52)(33,44,51,35)(40,54,55,53)(46,56,50,49);(10,14)(15,21)(16,22)(23,24)(27,39)(28,29)(32,48)(33,51)(34,41)(40,55)(42,43)(49,56)

Polycyclic presentation:

2^9|a^2=i,b^2=f,c^2=fhi,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=efh,d^c=dfh,e^c=ei,g^c=gi,e^d=ei

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6861337);