#348270

order	512
nilpotency class	3
isoclinism	isoclinic class of #346882
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
prime radical	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1059 conjugacy classes of subgroups; only the 160 normal subgroups are shown.

normal subgroups only

This group has 1059 conjugacy classes of subgroups — too many to draw. Only its 160 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) = 128

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

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

Polycyclic presentation:

2^9|a^2=i,b^2=eh,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dg,f^a=fh,g^a=ghi,f^c=fh,g^c=gi,f^d=fi,g^d=ghi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255466);