#187613

order	512
nilpotency class	3
isoclinism	isoclinic class of #18992
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2266 conjugacy classes of subgroups; only the 322 normal subgroups are shown.

normal subgroups only

This group has 2266 conjugacy classes of subgroups — too many to draw. Only its 322 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 — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,e^2=i,f^2=h,g^2=h,b^a=be,c^a=cf,d^a=dh,e^a=eh,e^b=ei,f^b=fhi,g^b=ghi,e^c=ehi,f^c=fh,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, 94809);