#233494

order	512
nilpotency class	3
isoclinism	isoclinic class of #233469
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂⁴	#127
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1280 conjugacy classes of subgroups; only the 222 normal subgroups are shown.

normal subgroups only

This group has 1280 conjugacy classes of subgroups — too many to draw. Only its 222 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) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,9,7)(8,11,10,12)(13,14,19,24)(15,25,29,35)(16,30,33,37)(17,23,18,22)(20,28,36,27)(21,32,38,31)(26,40,43,34)(39,42,44,41);(2,4)(6,11)(7,12)(8,10)(13,15,19,29)(14,20,24,36)(16,26,33,43)(17,28,18,27)(21,34,38,40)(22,35,23,25)(30,44,37,39)(31,42,32,41);(5,7)(6,9)(8,12)(10,11)(13,16,19,33)(14,21,24,38)(15,26,29,43)(17,31,18,32)(20,34,36,40)(22,30,23,37)(25,39,35,44)(27,41,28,42);(5,8,9,10)(6,11,7,12)(13,17,19,18)(14,22,24,23)(15,27,29,28)(16,31,33,32)(20,25,36,35)(21,30,38,37)(26,41,43,42)(34,39,40,44)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=i,d^2=h,e^2=h,f^2=i,g^2=hi,b^a=be,c^a=cf,d^a=di,e^a=eh,f^a=fi,g^a=ghi,c^b=cg,d^b=dh,e^b=eh,f^b=fi,g^b=ghi,e^c=eh,f^c=fi,g^c=ghi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 140690);