#262304

order	512
nilpotency class	3
isoclinism	isoclinic class of #246799
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#127	C₂³
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

860 conjugacy classes of subgroups; only the 152 normal subgroups are shown.

normal subgroups only

This group has 860 conjugacy classes of subgroups — too many to draw. Only its 152 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)(7,15)(8,12)(9,14)(10,13)(11,18)(16,20)(17,19)(21,22,25,29)(23,26,31,36)(24,27,33,28)(30,34,32,35);(1,2,4,3)(5,7)(6,11)(8,17)(9,18)(10,16)(12,20)(13,19)(14,15)(21,23,25,31)(22,26,29,36)(24,32,33,30)(27,35,28,34);(6,12)(7,16)(8,10)(13,14)(17,18)(19,20)(22,27)(24,33)(26,34)(28,29)(30,32)(35,36);(1,3,4,2)(5,8,9,10)(6,13,14,12)(7,17,18,16)(11,19,15,20)

Polycyclic presentation:

2^9|a^2=i,b^2=eh,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,f^c=fh,g^c=ghi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 169500);