#244863

order	512
nilpotency class	3
isoclinism	isoclinic class of #244791
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

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

1274 conjugacy classes of subgroups; only the 186 normal subgroups are shown.

normal subgroups only

This group has 1274 conjugacy classes of subgroups — too many to draw. Only its 186 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 152059);