#98069

order	512
nilpotency class	3
isoclinism	isoclinic class of #97214
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	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂⁴	C₂×C₂²⋊C₄
LCS	G ▷ C₂³×C₄ ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂³ ▷ C₁

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

normal subgroups only

This group has 530 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) = 40
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 3 generators on 40 points:

(5,8)(9,10,13,18,14,17,12,16)(11,19,20,24,21,15,22,23)(25,26,31,33,30,27,29,35)(28,32,38,39,37,34,36,40);(1,2)(3,5)(4,6)(7,8)(9,11,14,21)(10,15,17,19)(12,20,13,22)(16,23,18,24)(25,27,30,26)(28,34,37,32)(29,33,31,35)(36,39,38,40);(1,3,4,7)(2,5,6,8)(9,12,14,13)(10,16,17,18)(11,20,21,22)(15,23,19,24)(25,28,30,37)(26,32,27,34)(29,38,31,36)(33,40,35,39)

Polycyclic presentation:

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

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5265);