#140028

order	512
nilpotency class	4
isoclinism	isoclinic class of #139962
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1889
commutator	C₂×C₂²⋊C₄	C₂²×C₄
Frattini, cosocle	C₂×#117	C₂³
prime radical	#1595	C₄
LCS	G ▷ C₂×C₂²⋊C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#117 ◁ G
LECS	G ▷ C₂×#117 ▷ C₂³ ▷ C₂ ▷ C₁

459 conjugacy classes of subgroups; only the 62 normal subgroups are shown.

normal subgroups only

This group has 459 conjugacy classes of subgroups — too many to draw. Only its 62 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=df,d^2=g,e^2=gi,f^2=gi,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,h^a=hi,e^b=ehi,d^c=dh,e^c=egi,f^c=fh,e^d=ei,f^e=fi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 47224);