#125444

order	512
nilpotency class	4
isoclinism	isoclinic class of #125409
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1530
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₂×#97	C₂³
prime radical	C₂⁴	#100
LCS	G ▷ C₂×C₄² ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂×#97 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

348 conjugacy classes of subgroups; only the 68 normal subgroups are shown.

normal subgroups only

This group has 348 conjugacy classes of subgroups — too many to draw. Only its 68 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) = 72
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 3 generators on 72 points:

(1,2)(3,7,8,5)(4,6)(9,10,13,19,14,20,36,51,15,21,35,52,29,18,47,48)(11,22,24,54,25,16,38,40,26,41,59,49,37,42,33,64)(12,27,30,50,31,44,63,70,32,17,34,45,55,46,65,71)(23,43,56,69,28,60,61,68,57,53,58,72,62,39,66,67);(1,3)(2,5)(4,8)(6,7)(9,11,15,26)(10,16,21,42)(12,28,32,62)(13,33,35,38)(14,37,29,25)(17,43,27,53)(18,41,20,22)(19,49,52,54)(23,55,57,31)(24,36,59,47)(30,56,34,58)(39,44,60,46)(40,48,64,51)(45,68,50,67)(61,63,66,65)(69,70,72,71);(9,12,29,55,15,32,14,31)(10,17,18,44,21,27,20,46)(11,23,37,62,26,57,25,28)(13,34,47,63,35,30,36,65)(16,39,22,53,42,60,41,43)(19,50,48,71,52,45,51,70)(24,58,33,61,59,56,38,66)(40,68,54,69,64,67,49,72)

Polycyclic presentation:

2^9|a^2=e,b^2=i,c^2=hi,d^2=g,e^2=h,f^2=i,h^2=i,b^a=bd,c^a=cg,d^a=df,g^a=gi,c^b=ch,d^b=dg,e^b=efgi,f^b=fi,h^b=hi,e^c=ei,e^d=ei

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, 32640);