#98451

order	512
nilpotency class	3
isoclinism	isoclinic class of #98302
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

479 conjugacy classes of subgroups; only the 86 normal subgroups are shown.

normal subgroups only

This group has 479 conjugacy classes of subgroups — too many to draw. Only its 86 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) = 56
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 3 generators on 56 points:

(1,2,4,3,5,7,8,6)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23)(25,26)(27,35)(28,39,41,32)(29,42)(30,34)(31,37)(33,44)(36,50,52,49)(38,48)(40,47,56,54)(43,46)(45,51,53,55);(1,3,5,6)(2,4,7,8)(9,11)(10,15)(12,20)(13,22)(14,21)(16,23)(17,19)(18,24)(25,27,30,38)(26,31,34,46)(28,36,41,52)(29,43,44,37)(32,45,39,53)(33,48,42,35)(40,55,56,51)(47,50,54,49);(1,2)(3,4)(5,7)(6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28)(26,32)(27,36)(29,40)(30,41)(31,45)(33,47)(34,39)(35,49)(37,51)(38,52)(42,54)(43,55)(44,56)(46,53)(48,50)

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=gh,d^2=h,e^2=i,f^2=ghi,b^a=bd,c^a=ce,d^a=dg,e^a=eh,d^b=dh,e^b=ei,f^b=fgh,d^c=di,e^c=ei,f^c=fhi

Conjugacy classes & character table

53 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5647);