#299208

order	512
nilpotency class	3
isoclinism	isoclinic class of #299195
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2433
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

865 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 865 conjugacy classes of subgroups — too many to draw. Only its 190 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,c^2=fi,d^2=e,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,f^b=fh,g^c=gh

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 206404);