#112267

order	512
nilpotency class	3
isoclinism	isoclinic class of #97030
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁵	C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

1048 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 1048 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,d^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,c^b=cf,d^b=di,g^b=ghi,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 19463);