#382343

order	512
nilpotency class	3
isoclinism	isoclinic class of #30236
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2644
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂³×D₄	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

715 conjugacy classes of subgroups; only the 202 normal subgroups are shown.

normal subgroups only

This group has 715 conjugacy classes of subgroups — too many to draw. Only its 202 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:

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

Polycyclic presentation:

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

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, 289539);