#298358

order	512
nilpotency class	3
isoclinism	isoclinic class of #268211
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1124 conjugacy classes of subgroups; only the 156 normal subgroups are shown.

normal subgroups only

This group has 1124 conjugacy classes of subgroups — too many to draw. Only its 156 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) = 256

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,c^2=f,d^2=e,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=de,f^a=fh,g^a=gi,c^b=cg,f^b=fi,g^b=ghi,d^c=dhi,g^c=gh

Conjugacy classes & character table

47 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 205554);