#97485

order	512
nilpotency class	3
isoclinism	isoclinic class of #97444
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂³×C₄

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	C₂×#537	C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

730 conjugacy classes of subgroups; only the 90 normal subgroups are shown.

normal subgroups only

This group has 730 conjugacy classes of subgroups — too many to draw. Only its 90 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)(3,7,8,5)(4,6)(9,10,12,16,13,14,20,22)(11,17,18,24,19,15,21,23)(25,26,30,35,31,36,29,34)(27,37,41,32,40,47,39,46)(28,33,42,48,43,49,44,50)(38,52,53,56,54,51,55,45);(1,3,4,8)(2,5,6,7)(10,14)(12,20)(15,17)(18,21)(25,27)(26,32)(28,38)(29,41)(30,39)(31,40)(33,45)(34,47)(35,37)(36,46)(42,55)(43,54)(44,53)(48,51)(49,56)(50,52);(9,11,13,19)(10,15,14,17)(12,21,20,18)(16,24,22,23)(25,28)(26,33)(27,38)(29,44)(30,42)(31,43)(32,45)(34,50)(35,48)(36,49)(37,51)(39,55)(40,54)(41,53)(46,56)(47,52)

Polycyclic presentation:

2^9|a^2=f,b^2=gi,c^2=i,d^2=h,f^2=hi,b^a=bd,c^a=ce,d^a=dg,e^a=ei,d^b=dh,e^b=eh,f^b=fgh,d^c=dh,f^c=fi

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, 4681);