#606854

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

Polycyclic presentation:

2^9|a^2=i,c^2=fgh,d^2=h,e^2=fh,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, 514050);