#496117

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #376 ◁ #1726 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

756 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 72 points:

(2,4)(5,6,8,7)(9,10,14,19)(11,21,25,45)(12,17,27,38)(13,30,32,58)(15,33,31,20)(16,34,36,24)(18,41,42,22)(23,43,49,65)(26,53,52,46)(28,56,54,71)(29,57,55,40)(35,59,61,50)(37,47,44,60)(39,64,63,48)(51,70,69,67)(62,68,66,72);(9,11,22,47,15,26,32,34,14,25,41,60,31,52,13,24)(10,16,30,46,20,37,42,45,19,36,58,53,33,44,18,21)(12,23,48,68,29,51,54,59,27,49,64,72,55,69,28,50)(17,35,56,67,40,62,63,65,38,61,71,70,57,66,39,43);(1,2,3,4)(9,12,14,27)(10,17,19,38)(11,23,25,49)(13,28,32,54)(15,29,31,55)(16,35,36,61)(18,39,42,63)(20,40,33,57)(21,43,45,65)(22,48,41,64)(24,50,34,59)(26,51,52,69)(30,56,58,71)(37,62,44,66)(46,67,53,70)(47,68,60,72);(5,7,8,6)(12,27)(17,38)(23,49)(28,54)(29,55)(35,61)(39,63)(40,57)(43,65)(48,64)(50,59)(51,69)(56,71)(62,66)(67,70)(68,72)

Polycyclic presentation:

2^9|a^2=g,b^2=eh,c^2=fi,d^2=gi,e^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,h^a=hi,d^c=di

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, 403313);