#111624

order	512
nilpotency class	3
isoclinism	isoclinic class of #99835
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#386
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

944 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 944 conjugacy classes of subgroups — too many to draw. Only its 150 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) = 44
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,d^2=i,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,f^a=fh,c^b=cf,d^b=di,g^b=ghi,d^c=dh,g^c=gh

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, 18820);