#117118

order	512
nilpotency class	3
isoclinism	isoclinic class of #10218
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#392
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
socle	C₂²	#1439
prime radical	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

596 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 596 conjugacy classes of subgroups — too many to draw. Only its 170 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)(17,18,22,26,23,28,21,27)(19,29,33,39,32,40,31,24)(20,25,35,43,36,44,37,42)(30,45,34,38,46,41,47,48);(2,6)(7,15)(8,13)(9,16)(17,19,21,31,23,32,22,33)(18,24,26,29,28,39,27,40)(20,34,35,46,36,47,37,30)(25,41,42,38,44,45,43,48);(1,3,4,10)(2,7,6,15)(5,14,12,11)(8,16,13,9)(17,20)(18,25)(19,30)(21,37)(22,35)(23,36)(24,38)(26,43)(27,42)(28,44)(29,41)(31,47)(32,46)(33,34)(39,48)(40,45)

Polycyclic presentation:

2^9|a^2=g,b^2=di,c^2=i,d^2=h,e^2=i,f^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,e^b=eh,d^c=dh,e^c=ei,f^c=fh

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, 24314);