#316205

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	C₂⁴	#124
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

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

normal subgroups only

This group has 415 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=eghi,b^2=e,c^2=efi,d^2=ei,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, 223401);