#495038

order	512
nilpotency class	4
isoclinism	isoclinic class of #2221
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#466
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#466
prime radical	C₂⁴	#135
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

628 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=fh,d^2=i,e^2=i,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fh,h^a=hi,e^b=ei

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402234);