#382603

order	512
nilpotency class	3
isoclinism	isoclinic class of #213736
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2308
commutator, prime radical	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1177 conjugacy classes of subgroups; only the 314 normal subgroups are shown.

normal subgroups only

This group has 1177 conjugacy classes of subgroups — too many to draw. Only its 314 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)(5,6)(7,11)(8,12)(9,14)(10,13)(15,17)(16,24)(18,25)(19,26)(20,28)(21,27)(22,30)(23,29)(31,33)(32,35)(34,36);(5,6)(7,15,17,11)(8,12)(9,14)(10,13)(16,26,19,24)(18,31,33,25)(20,28)(21,27)(22,30)(23,29)(32,36,34,35);(2,4)(5,7,9,18,10,17,23,33)(6,11,14,25,13,15,29,31)(8,19,20,34,21,16,22,32)(12,26,28,36,27,24,30,35);(5,8,10,21)(6,12,13,27)(7,16,17,19)(9,22,23,20)(11,24,15,26)(14,30,29,28)(18,34,33,32)(25,36,31,35)

Polycyclic presentation:

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

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, 289799);