#134993

order	512
nilpotency class	4
isoclinism	isoclinic class of #134945
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1807
commutator	C₄×C₈	C₂²×C₄
Frattini, cosocle	C₂×C₄×C₈	C₂³
prime radical	C₂³	#465
LCS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ #2960 ◁ G
LECS	G ▷ C₂×C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

245 conjugacy classes of subgroups; only the 98 normal subgroups are shown.

normal subgroups only

This group has 245 conjugacy classes of subgroups — too many to draw. Only its 98 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)(3,9)(4,6)(5,13)(7,11)(8,12)(10,16)(14,15)(17,18)(19,30,34,48)(20,35,37,45)(21,31)(22,27)(23,29)(24,32,38,25)(26,39)(28,41)(33,47,42,36)(40,43)(44,46);(2,6)(7,15)(8,13)(9,16)(17,19,21,32,22,33,39,45,23,34,40,25,41,42,46,35)(18,24,26,30,28,20,31,36,29,38,44,48,27,37,43,47);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20)(18,25)(19,31)(21,36)(22,38)(23,37)(24,41)(26,42)(27,45)(28,35)(29,32)(30,46)(33,44)(34,43)(39,48)(40,47)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 42189);