#136998

order	512
nilpotency class	4
isoclinism	isoclinic class of #12316
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#447
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

363 conjugacy classes of subgroups; only the 112 normal subgroups are shown.

normal subgroups only

This group has 363 conjugacy classes of subgroups — too many to draw. Only its 112 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 — 3 generators on 52 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=dfghi,c^2=hi,d^2=gi,e^2=h,f^2=gi,g^2=i,b^a=bd,c^a=ce,d^b=dg,e^b=eh,f^b=fg,g^b=gi,d^c=dh,e^c=eh,f^c=fh

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 44194);