#249083

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	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1542 conjugacy classes of subgroups; only the 184 normal subgroups are shown.

normal subgroups only

This group has 1542 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)(14,19)(16,23)(17,24)(18,25)(20,30)(21,31)(22,32)(26,36)(27,37)(28,38)(29,39)(33,42)(34,35)(40,44)(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);(13,17,18,29)(14,21,22,28)(15,24,25,39)(16,26,27,40)(19,31,32,38)(20,33,34,41)(23,36,37,44)(30,42,35,43)

Polycyclic presentation:

2^9|a^2=h,d^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, 156279);