#110932

order	512
nilpotency class	3
isoclinism	isoclinic class of #110922
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1437
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

762 conjugacy classes of subgroups; only the 194 normal subgroups are shown.

normal subgroups only

This group has 762 conjugacy classes of subgroups — too many to draw. Only its 194 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,d^2=hi,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,c^b=ch,d^b=di,e^b=ei,f^b=fi,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 18128);