#97034

order	512
nilpotency class	3
isoclinism	isoclinic class of #97030
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

702 conjugacy classes of subgroups; only the 102 normal subgroups are shown.

normal subgroups only

This group has 702 conjugacy classes of subgroups — too many to draw. Only its 102 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=g,d^2=h,f^2=i,b^a=bd,c^a=ce,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^b=fgh,f^c=fh

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4230);