#332984

order	512
nilpotency class	3
isoclinism	isoclinic class of #332983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3496	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

788 conjugacy classes of subgroups; only the 138 normal subgroups are shown.

normal subgroups only

This group has 788 conjugacy classes of subgroups — too many to draw. Only its 138 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=defh,f^a=fi,c^b=cg,d^b=de,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

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, 240180);