#249260

order	512
nilpotency class	3
isoclinism	isoclinic class of #249259
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	#36967	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1365 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1365 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:

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

Polycyclic presentation:

2^9|a^2=h,f^2=i,b^a=be,c^a=cf,d^a=deh,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

50 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 156456);