#321985

order	512
nilpotency class	3
isoclinism	isoclinic class of #321983
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2541
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#3147	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

720 conjugacy classes of subgroups; only the 154 normal subgroups are shown.

normal subgroups only

This group has 720 conjugacy classes of subgroups — too many to draw. Only its 154 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:

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

Polycyclic presentation:

2^9|a^2=fh,b^2=egh,d^2=e,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dei,e^a=eh,c^b=cg,d^b=dh,f^c=fi,g^c=gh

Conjugacy classes & character table

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 229181);