#352850

order	512
nilpotency class	3
isoclinism	isoclinic class of #352064
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂×C₄×C₈

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#127
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#2867	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

724 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 724 conjugacy classes of subgroups — too many to draw. Only its 170 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) = 256

A minimal faithful permutation representation — 4 generators on 40 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=gh,c^2=fh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dg,e^a=eh,f^a=fi,g^a=gh,c^b=ci,d^b=dhi

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, 260046);