#249049

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	C₂²×#121	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1499 conjugacy classes of subgroups; only the 170 normal subgroups are shown.

normal subgroups only

This group has 1499 conjugacy classes of subgroups — too many to draw. Only its 170 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=hi,c^2=i,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, 156245);