#122153

order	512
nilpotency class	3
isoclinism	isoclinic class of #97710
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

580 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 56 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=f,c^2=f,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,e^b=eh,g^b=ghi,d^c=dh,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, 29349);