#116358

order	512
nilpotency class	3
isoclinism	isoclinic class of #10390
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#393
commutator	C₂⁵	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
socle	C₂²	#1439
prime radical	C₂³×#41	C₂²
LCS	G ▷ C₂⁵ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

1424 conjugacy classes of subgroups; only the 130 normal subgroups are shown.

normal subgroups only

This group has 1424 conjugacy classes of subgroups — too many to draw. Only its 130 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) = 64

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=d,g^2=h,b^a=bd,c^a=ce,e^a=eh,f^a=fhi,c^b=cf,e^b=ei,f^b=fh,d^c=dh,g^c=gh

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, 23554);