#121816

order	512
nilpotency class	3
isoclinism	isoclinic class of #121813
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

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

normal subgroups only

This group has 407 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=de,c^2=de,d^2=h,e^2=hi,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,f^a=fi,c^b=cf,d^b=di,e^b=ei,f^b=fh,f^c=fh,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, 29012);