#347158

order	512
nilpotency class	3
isoclinism	isoclinic class of #347154
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

2182 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2182 conjugacy classes of subgroups — too many to draw. Only its 188 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 — 4 generators on 40 points:

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

Polycyclic presentation:

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

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, 254354);