#391865

order	512
nilpotency class	3
isoclinism	isoclinic class of #232147
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2380
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#532	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1336 conjugacy classes of subgroups; only the 234 normal subgroups are shown.

normal subgroups only

This group has 1336 conjugacy classes of subgroups — too many to draw. Only its 234 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=h,c^2=i,d^2=e,f^2=i,b^a=be,c^a=cf,h^a=hi,c^b=cg,d^b=def,e^b=ei,d^c=de,f^c=fi

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, 299061);