#391530

order	512
nilpotency class	3
isoclinism	isoclinic class of #32875
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2842
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂²×Q₈
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

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

normal subgroups only

This group has 741 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) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 298726);