#98432

order	512
nilpotency class	3
isoclinism	isoclinic class of #98302
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	#379
commutator	C₂×C₄²	C₂²×C₄
Frattini, cosocle	C₄³	C₂³
prime radical	C₂²×D₄	C₂²⋊C₄
LCS	G ▷ C₂×C₄² ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₄³ ▷ C₂³ ▷ C₁

608 conjugacy classes of subgroups; only the 100 normal subgroups are shown.

normal subgroups only

This group has 608 conjugacy classes of subgroups — too many to draw. Only its 100 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)(3,6)(4,7)(5,8)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23)(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)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(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);(2,7)(3,4)(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)(31,44)(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=gh,c^2=gi,d^2=h,e^2=i,f^2=gi,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, 5628);