#392235

order	512
nilpotency class	3
isoclinism	isoclinic class of #190022
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#381
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3358 conjugacy classes of subgroups; only the 438 normal subgroups are shown.

normal subgroups only

This group has 3358 conjugacy classes of subgroups — too many to draw. Only its 438 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299431);