#108261

order	512
nilpotency class	3
isoclinism	isoclinic class of #108238
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	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	C₂×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 852 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)(17,18,21,26,23,27,22,36)(19,28,32,24,31,39,30,38)(20,25,34,41,35,42,33,40)(29,44,45,48,46,43,47,37);(1,3)(2,6)(4,10)(5,11)(7,13)(8,9)(12,16)(14,15)(17,19,22,32,23,31,21,30)(18,24,26,28,27,38,36,39)(20,29,33,45,35,46,34,47)(25,37,41,43,42,48,40,44);(1,4,5,12)(2,7,8,14)(3,10,11,16)(6,13,9,15)(17,20)(18,25)(19,29)(21,34)(22,33)(23,35)(24,37)(26,41)(27,42)(28,43)(30,47)(31,46)(32,45)(36,40)(38,48)(39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=i,f^2=i,g^2=i,b^a=bd,c^a=ce,d^a=dh,g^a=gi,e^b=ei,f^b=fhi,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, 15457);