#606851

order	512
nilpotency class	2
isoclinism	isoclinic class of #53158
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, socle, cosocle	C₂⁴	C₂⁵
commutator	C₂³	C₂⁴×C₄
prime radical	C₂⁵	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1676 conjugacy classes of subgroups; only the 746 normal subgroups are shown.

normal subgroups only

This group has 1676 conjugacy classes of subgroups — too many to draw. Only its 746 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) = 256

A minimal faithful permutation representation — 5 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,c^2=f,d^2=fg,e^2=fgh,b^a=bf,c^a=cg,d^a=dfh,e^a=efg,c^b=ch,d^b=dg,e^b=ef

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 514047);