#249058

order	512
nilpotency class	3
isoclinism	isoclinic class of #249019
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1239 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1239 conjugacy classes of subgroups — too many to draw. Only its 162 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156254);