#497081

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
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

496 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 496 conjugacy classes of subgroups — too many to draw. Only its 234 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)(21,22,25,30,26,31,38,44,27,23,28,33,39,34,41,47)(24,29,36,42,37,43,49,51,35,32,40,45,48,46,50,52);(5,7)(6,11)(8,17)(9,15)(10,16)(12,20)(13,14)(18,19)(21,23,25,33,26,34,38,47,27,22,28,30,39,31,41,44)(24,32,36,45,37,46,49,52,35,29,40,42,48,43,50,51);(1,2,3,4)(5,8,10,9)(6,12,13,18)(7,15,16,17)(11,19,14,20)(21,24)(22,29)(23,32)(25,36)(26,37)(27,35)(28,40)(30,42)(31,43)(33,45)(34,46)(38,49)(39,48)(41,50)(44,51)(47,52);(5,9,10,8)(6,12,13,18)(7,15,16,17)(11,20,14,19)(24,35)(29,32)(36,40)(37,48)(42,45)(43,46)(49,50)(51,52)

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=de,c^b=ce,d^c=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 404277);