#348092

order	512
nilpotency class	3
isoclinism	isoclinic class of #348090
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

883 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 883 conjugacy classes of subgroups — too many to draw. Only its 152 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) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|b^2=eh,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ch,f^c=fi,g^c=ghi,f^d=fhi,g^d=gh

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, 255288);