#347155

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

2308 conjugacy classes of subgroups; only the 188 normal subgroups are shown.

normal subgroups only

This group has 2308 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)(9,10)(11,16)(12,17)(13,15)(14,19)(18,25)(20,33)(21,26)(22,30)(23,27)(24,32)(28,35)(29,36)(31,37)(34,39)(38,40);(3,7)(5,8)(10,15)(13,25)(16,26)(17,27)(18,19)(21,36)(23,37)(28,38)(29,30)(31,32)(33,39)(34,40);(1,2)(3,8)(4,6)(5,7)(9,11)(10,16)(12,20)(13,21)(14,22)(15,26)(17,28)(18,29)(19,30)(23,34)(24,35)(25,36)(27,38)(31,39)(32,33)(37,40);(9,12,14,24)(10,17,19,32)(11,20,22,35)(13,23,25,37)(15,27,18,31)(16,28,30,33)(21,34,36,40)(26,38,29,39)

Polycyclic presentation:

2^9|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, 254351);