#141485

order	512
nilpotency class	4
isoclinism	isoclinic class of #141481
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1964
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×C₄×Q₈	C₂³
prime radical	C₂³	#390
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×C₈×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

322 conjugacy classes of subgroups; only the 78 normal subgroups are shown.

normal subgroups only

This group has 322 conjugacy classes of subgroups — too many to draw. Only its 78 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) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 32 points:

(1,2)(3,11)(4,7)(5,14)(6,10)(8,12)(9,15)(13,16)(17,18,23,24,22,26,20,31)(19,27,29,28,30,21,25,32);(1,3,5,4,6,12,15,13)(2,7,9,11,10,16,14,8)(17,19,28,18,22,30,32,26)(20,29,21,31,23,25,27,24);(1,4,5,12,6,13,15,3)(2,8,9,7,10,11,14,16)(17,20,22,23)(18,24,26,31)(19,29,30,25)(21,28,27,32)

Polycyclic presentation:

2^9|a^2=f,b^2=dfi,c^2=e,d^2=gh,e^2=ghi,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eghi,g^a=gi,h^a=hi,c^b=cg,d^b=dh,f^b=fh,g^b=gi,h^b=hi,d^c=di,f^d=fi

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 48681);