#249047

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	C₂×#392	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1430 conjugacy classes of subgroups; only the 162 normal subgroups are shown.

normal subgroups only

This group has 1430 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) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=i,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, 156243);