#606852

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

Polycyclic presentation:

2^9|a^2=i,c^2=fh,d^2=f,e^2=gh,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, 514048);