#299240

order	512
nilpotency class	3
isoclinism	isoclinic class of #299233
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1022 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 1022 conjugacy classes of subgroups — too many to draw. Only its 154 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) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 48 points:

(3,9)(4,10)(6,12)(8,13)(11,15)(14,16)(17,18)(19,27)(20,32)(21,36)(22,26)(23,30)(24,34)(25,33)(28,41)(29,42)(31,40)(35,45)(37,44)(38,46)(39,43)(47,48);(1,2)(3,6)(4,8)(5,7)(9,12)(10,13)(11,14)(15,16)(17,19,21,30,22,31,33,41)(18,23,25,27,26,28,36,40)(20,29,34,43,35,44,46,48)(24,37,32,39,38,42,45,47);(1,3,10,15,5,11,4,9)(2,6,13,16,7,14,8,12)(17,18)(19,28,31,23)(20,24)(21,25)(22,26)(27,30,40,41)(29,42,44,37)(32,34)(33,36)(35,38)(39,43,47,48)(45,46);(2,7)(6,14)(8,13)(12,16)(17,20,33,46,22,35,21,34)(18,24,36,45,26,38,25,32)(19,29,41,48,31,44,30,43)(23,37,40,47,28,42,27,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206436);