#493130

order	512
nilpotency class	2
isoclinism	isoclinic class of #7855
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×C₄²
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂²×#97
prime radical	C₂⁴	C₂×M₁₆
LCS	G ▷ C₂×C₄ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

728 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)(5,6,10,15,11,17,29,27,12,16,30,26,9,14,25,8)(7,18,22,35,23,13,31,19,20,33,28,24,21,32,36,34);(2,4)(5,7)(6,13,16,32)(8,24,27,35)(9,21)(10,28)(11,23)(12,20)(14,18,17,33)(15,34,26,19)(22,30)(25,31)(29,36);(5,8,25,14,9,26,30,16,12,27,29,17,11,15,10,6)(7,19,36,13,21,35,28,18,20,34,31,32,23,24,22,33);(7,20)(13,32)(18,33)(19,34)(21,23)(22,28)(24,35)(31,36)

Polycyclic presentation:

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

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313257, #72313284, #72313298, …
maximal normal subgroup of	#72313257, #72313284, #72313298, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400326);