#132106

order	512
nilpotency class	4
isoclinism	isoclinic class of #131985
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1781
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₂²⋊C₄	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂²×#123 ◁ G
LECS	G ▷ C₂²×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁

794 conjugacy classes of subgroups; only the 84 normal subgroups are shown.

normal subgroups only

This group has 794 conjugacy classes of subgroups — too many to draw. Only its 84 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)(3,9,11,16)(4,12,14,7)(5,8)(6,10)(13,15)(17,18,22,27,23,28,21,26)(19,29,33,24,32,39,31,38)(20,25,34,40,35,41,36,42)(30,44,45,48,46,43,47,37);(1,3,5,11)(2,6,8,15)(4,13,14,10)(7,16,12,9)(17,19)(18,24)(20,30)(21,33)(22,31)(23,32)(25,37)(26,39)(27,29)(28,38)(34,47)(35,46)(36,45)(40,43)(41,48)(42,44);(17,20)(18,25)(19,30)(21,36)(22,34)(23,35)(24,37)(26,42)(27,40)(28,41)(29,43)(31,47)(32,46)(33,45)(38,48)(39,44)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,g^a=gi,d^b=dh,e^b=ehi,f^b=fgh,d^c=dhi,f^d=fi

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72227718, #72227721, #72227724
maximal normal subgroup of	#72227718, #72227721, #72227724

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 39302);