#93509

order	512
nilpotency class	3
isoclinism	isoclinic class of #93508
automorphism	?
homology	Abelianization (H₁) = C₄² · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	H(3,ℤ/4)
commutator	C₂²×C₈	C₄²
Frattini, cosocle	C₂²×#131	C₂²
socle	C₂²	#1289
prime radical	C₂²×#41	D₄
LCS	G ▷ C₂²×C₈ ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂²×C₈ ◁ G
LECS	G ▷ C₂²×#131 ▷ C₂²×C₄ ▷ C₂ ▷ C₁

462 conjugacy classes of subgroups; only the 72 normal subgroups are shown.

normal subgroups only

This group has 462 conjugacy classes of subgroups — too many to draw. Only its 72 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 — 2 generators on 72 points:

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

Polycyclic presentation:

2^9|a^2=d,b^2=e,c^2=fhi,d^2=i,f^2=i,g^2=i,b^a=bc,c^a=cf,e^a=efghi,h^a=hi,c^b=cg,d^b=dh,h^b=hi,d^c=di,e^c=ei,e^d=ei

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 705);