#497099

order	512
nilpotency class	4
isoclinism	isoclinic class of #33764
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×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

474 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 474 conjugacy classes of subgroups — too many to draw. Only its 194 normal subgroups are shown.

Dotted edges: containments with no intermediate normal subgroup — omitted non-normal subgroups may lie between.

Full subgroup data →

Presentation

minimal degree	μ(G) = 52
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 52 points:

(1,2,4,3)(5,6)(7,14)(8,12)(9,18)(10,13)(11,16)(15,20)(17,19)(21,22)(23,32)(24,28)(25,38)(26,31)(27,35)(29,33)(30,37)(34,45)(36,47)(39,43)(40,48)(41,52)(42,51)(44,49)(46,50);(1,3,4,2)(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,22)(23,27)(24,30)(25,29)(26,31)(28,37)(32,35)(33,38)(34,41)(36,42)(39,44)(40,50)(43,49)(45,52)(46,48)(47,51);(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,23,33,48,24,34,39,51,26,36,49,52,37,50,25,35)(22,27,38,46,30,41,44,47,31,42,43,45,28,40,29,32);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(21,24,26,37)(22,28,31,30)(23,34,36,50)(25,33,39,49)(27,40,42,41)(29,43,44,38)(32,45,47,46)(35,48,51,52)

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=efh,d^2=ei,f^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,f^a=fh,h^a=hi,c^b=ce,d^b=di

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404295);