#147631

order	512
nilpotency class	4
isoclinism	isoclinic class of #129559
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1786
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1012 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54827);