#127801

order	512
nilpotency class	4
isoclinism	isoclinic class of #11692
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#99
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂×C₄×Q₈	C₂³
socle	C₂²	C₂×#342
prime radical	C₂⁴	#95
LCS	G ▷ C₂²×C₄ ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ C₂²×C₄×Q₈ ◁ G
LECS	G ▷ C₂×C₄×Q₈ ▷ C₂³ ▷ C₂ ▷ C₁

685 conjugacy classes of subgroups; only the 148 normal subgroups are shown.

normal subgroups only

This group has 685 conjugacy classes of subgroups — too many to draw. Only its 148 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) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 40 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 34997);