#347861

order	512
nilpotency class	3
isoclinism	isoclinic class of #347788
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#121
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

847 conjugacy classes of subgroups; only the 174 normal subgroups are shown.

normal subgroups only

This group has 847 conjugacy classes of subgroups — too many to draw. Only its 174 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 — 4 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,c^2=i,e^2=h,f^2=i,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,c^b=ch,d^b=di,d^c=dh,f^c=fi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255057);