#244115

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	#10032	C₂
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

698 conjugacy classes of subgroups; only the 150 normal subgroups are shown.

normal subgroups only

This group has 698 conjugacy classes of subgroups — too many to draw. Only its 150 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:

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

Polycyclic presentation:

2^9|a^2=eh,b^2=h,d^2=ef,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, 151311);