#147707

order	512
nilpotency class	4
isoclinism	isoclinic class of #13313
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
socle	C₂²	#1893
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1010 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 68
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 68 points:

(1,2)(3,4)(5,6)(7,18,22,49)(8,23,27,47)(9,29)(10,15)(11,17)(12,20,38,58)(13,26,42,28)(14,32)(16,35)(19,51)(21,43,34,50)(24,60)(25,44,31,41)(30,37,57,39)(33,45)(36,65)(40,52)(46,55)(48,53)(54,68)(56,67)(59,62,64,63)(61,66);(5,7)(6,12)(8,24)(9,30)(10,34)(11,22)(13,40)(14,43)(15,39)(16,37)(17,38)(18,45)(19,26)(20,55)(21,35)(23,36)(25,56)(27,53)(28,54)(29,49)(31,52)(32,58)(33,57)(41,60)(42,67)(44,48)(46,50)(47,66)(51,59)(61,62)(63,65)(64,68);(2,4)(5,8)(6,13)(7,19)(9,31)(10,28)(11,27)(12,36)(14,44)(15,47)(16,23)(17,42)(18,48)(20,56)(21,53)(22,54)(24,34)(25,33)(26,35)(29,64)(30,65)(32,62)(37,67)(38,66)(39,40)(41,46)(43,68)(45,59)(49,60)(50,51)(52,58)(55,63)(57,61)

Polycyclic presentation:

2^9|a^2=g,d^2=h,e^2=i,f^2=i,h^2=i,b^a=bd,c^a=ce,d^a=dh,h^a=hi,c^b=cf,d^b=dhi,e^b=ei,f^b=fi,g^b=gi,h^b=hi,d^c=di,e^c=ei,f^c=fi,g^c=gi

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54903);