#249081

order	512
nilpotency class	3
isoclinism	isoclinic class of #249079
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1564 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1564 conjugacy classes of subgroups — too many to draw. Only its 184 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,f^a=fi,c^b=cg,f^b=fh,e^c=eh,f^c=fi,f^d=fh

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, 156277);