#238531

order	512
nilpotency class	3
isoclinism	isoclinic class of #232291
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	C₂³×C₄	C₂⁴
prime radical	#2656	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1070 conjugacy classes of subgroups; only the 190 normal subgroups are shown.

normal subgroups only

This group has 1070 conjugacy classes of subgroups — too many to draw. Only its 190 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 145727);