#493110

order	512
nilpotency class	3
isoclinism	isoclinic class of #20671
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#117
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×#432
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

766 conjugacy classes of subgroups.

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,g^2=h,h^2=i,b^a=be,g^a=gi,c^b=ch,e^b=ei,d^c=di,g^c=gi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226782, #72226795
maximal normal subgroup of	#72226782, #72226795

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400306);