#249211

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	C₂×#537	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1272 conjugacy classes of subgroups; only the 166 normal subgroups are shown.

normal subgroups only

This group has 1272 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=h,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, 156407);