#135624

order	512
nilpotency class	4
isoclinism	isoclinic class of #135590
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1823
commutator	C₂²×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	C₂×#102
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

267 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 267 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

65 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42820);