#411852

order	512
nilpotency class	3
isoclinism	isoclinic class of #395246
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3067	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1980 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 4 generators on 68 points:

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

Polycyclic presentation:

2^9|b^2=e,d^2=hi,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,f^c=fi,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, 319048);