#333111

order	512
nilpotency class	3
isoclinism	isoclinic class of #333055
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂²×D₄	C₂×D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

673 conjugacy classes of subgroups; only the 132 normal subgroups are shown.

normal subgroups only

This group has 673 conjugacy classes of subgroups — too many to draw. Only its 132 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:

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

Polycyclic presentation:

2^9|a^2=eh,c^2=hi,d^2=i,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=dei,e^b=eh,g^b=gh,f^c=fi,g^c=gh,e^d=eh,f^d=fi

Conjugacy classes & character table

44 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 240307);