#392707

order	512
nilpotency class	3
isoclinism	isoclinic class of #32894
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2846
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3719 conjugacy classes of subgroups; only the 418 normal subgroups are shown.

normal subgroups only

This group has 3719 conjugacy classes of subgroups — too many to draw. Only its 418 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=eh,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dg,e^b=ei,h^b=hi,g^c=gi,f^d=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, 299903);