#238654

order	512
nilpotency class	3
isoclinism	isoclinic class of #233668
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

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

920 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 920 conjugacy classes of subgroups — too many to draw. Only its 186 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) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,d^2=h,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,d^b=dh,f^b=fi,g^b=ghi,d^c=dhi,f^c=fh,g^c=ghi

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, 145850);