#121807

order	512
nilpotency class	3
isoclinism	isoclinic class of #121806
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1445
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	#123
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

413 conjugacy classes of subgroups; only the 116 normal subgroups are shown.

normal subgroups only

This group has 413 conjugacy classes of subgroups — too many to draw. Only its 116 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,g^c=gh

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, 29003);