#107126

order	512
nilpotency class	3
isoclinism	isoclinic class of #107114
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂⁴	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

914 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 914 conjugacy classes of subgroups — too many to draw. Only its 186 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)(3,9,11,6)(4,7)(5,8)(10,15,16,13)(12,14)(17,18,21,25,22,26,33,39)(19,27,30,41,29,23,34,36)(20,24,32,38,31,37,44,47)(28,35,43,46,42,40,45,48);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19)(18,23,26,27)(20,28)(21,30)(22,29)(24,35,37,40)(25,36,39,41)(31,42)(32,43)(33,34)(38,46,47,48)(44,45);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,24)(19,28)(21,32)(22,31)(23,35)(25,38)(26,37)(27,40)(29,42)(30,43)(33,44)(34,45)(36,46)(39,47)(41,48)

Polycyclic presentation:

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

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, 14322);