#105925

order	512
nilpotency class	3
isoclinism	isoclinic class of #105909
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂²⋊Q₈
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

739 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 739 conjugacy classes of subgroups — too many to draw. Only its 188 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=h,d^2=h,e^2=i,f^2=h,g^2=hi,b^a=bd,c^a=ce,d^a=dh,g^a=ghi,c^b=ch,d^b=di,e^b=ei,d^c=di,e^c=ei,f^c=fi

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, 13121);