#318091

order	512
nilpotency class	3
isoclinism	isoclinic class of #288755
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle, prime radical	C₂³	D₄²
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

600 conjugacy classes of subgroups; only the 158 normal subgroups are shown.

normal subgroups only

This group has 600 conjugacy classes of subgroups — too many to draw. Only its 158 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) = 36
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eghi,c^2=i,d^2=ehi,e^2=h,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,d^c=dh,g^c=gi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 225287);