#348745

order	512
nilpotency class	3
isoclinism	isoclinic class of #346582
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×#555	C₂²
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1025 conjugacy classes of subgroups; only the 180 normal subgroups are shown.

normal subgroups only

This group has 1025 conjugacy classes of subgroups — too many to draw. Only its 180 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

77 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 255941);