#316200

order	512
nilpotency class	3
isoclinism	isoclinic class of #316187
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#29810	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

499 conjugacy classes of subgroups; only the 122 normal subgroups are shown.

normal subgroups only

This group has 499 conjugacy classes of subgroups — too many to draw. Only its 122 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:

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

Polycyclic presentation:

2^9|a^2=eghi,b^2=e,c^2=ef,d^2=ehi,e^2=h,f^2=hi,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, 223396);