#497085

order	512
nilpotency class	2
isoclinism	isoclinic class of #2812
automorphism	?
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

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

468 conjugacy classes of subgroups; only the 226 normal subgroups are shown.

normal subgroups only

This group has 468 conjugacy classes of subgroups — too many to draw. Only its 226 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=g,c^2=ef,d^2=e,g^2=h,h^2=i,b^a=be,c^a=cf,d^a=dei,c^b=cei,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, 404281);