#495216

order	512
nilpotency class	4
isoclinism	isoclinic class of #33607
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂²×D₈
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂⁴	C₂×D₈
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#117 ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

1078 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 72
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=fh,d^2=i,e^2=hi,f^2=hi,h^2=i,b^a=be,c^a=cf,e^a=eh,f^a=fh,h^a=hi,d^b=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, 402412);