#134178

order	512
nilpotency class	4
isoclinism	isoclinic class of #133903
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1782
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #2296 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

498 conjugacy classes of subgroups; only the 82 normal subgroups are shown.

normal subgroups only

This group has 498 conjugacy classes of subgroups — too many to draw. Only its 82 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=i,f^2=g,b^a=bd,c^a=ce,d^a=dg,e^a=eh,h^a=hi,c^b=ci,f^b=fgi,f^c=fh,f^e=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72293050, #72293070, #72293072
maximal normal subgroup of	#72293050, #72293070, #72293072

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 41374);