#238578

order	512
nilpotency class	3
isoclinism	isoclinic class of #233754
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

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

normal subgroups only

This group has 934 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)(15,19)(16,26)(17,29)(18,22)(20,28)(21,32)(23,40)(24,38)(25,35)(27,39)(30,34)(31,37)(33,42)(36,43)(41,44);(1,2)(3,4)(5,7,10,6)(8,11,9,12)(13,15,18,25)(14,19,22,35)(16,23,27,41)(17,30,32,24)(20,36,37,33)(21,38,29,34)(26,40,39,44)(28,43,31,42);(1,3,4,2)(6,11)(7,12)(8,9)(13,16)(14,20)(15,23)(17,31)(18,27)(19,33)(21,39)(22,37)(24,43)(25,41)(26,29)(28,32)(30,42)(34,44)(35,36)(38,40);(16,27)(20,37)(23,41)(26,39)(28,31)(33,36)(40,44)(42,43)

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=di,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, 145774);