#244121

order	512
nilpotency class	3
isoclinism	isoclinic class of #10012
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂²	#2397
prime radical	#3150	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

676 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 676 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 151317);