#461206

order	512
nilpotency class	3
isoclinism	isoclinic class of #461203
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

826 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=f,c^2=i,d^2=ei,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,c^b=cg,g^b=gi,d^c=de,e^c=ei,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 368402);