#6436383

order	512
nilpotency class	3
isoclinism	isoclinic class of #6433042
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂×#376 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

2246 conjugacy classes of subgroups; only the 618 normal subgroups are shown.

normal subgroups only

This group has 2246 conjugacy classes of subgroups — too many to draw. Only its 618 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 — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=f,c^2=g,d^2=hi,e^2=i,f^2=i,b^a=bf,c^a=cg,d^a=dh,f^a=fi,c^b=ch,e^b=ei,d^c=di

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6343579);