#249234

order	512
nilpotency class	3
isoclinism	isoclinic class of #249199
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1343 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1343 conjugacy classes of subgroups — too many to draw. Only its 166 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156430);