#493140

order	512
nilpotency class	4
isoclinism	isoclinic class of #509
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂×D₈
commutator	C₈	C₂²×C₄²
Frattini, cosocle	C₂²×C₈	C₂⁴
socle	C₂²	C₂×C₄×D₈
prime radical	C₂⁴	#135
LCS	G ▷ C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂³×C₈ ◁ C₂²×C₄×C₈ ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

740 conjugacy classes of subgroups.

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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=efh,e^2=hi,f^2=h,h^2=i,b^a=be,d^a=di,e^b=eh,f^b=fh,h^b=hi

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72210510
maximal normal subgroup of	#72210510
maximal quotient of	#72210412

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 400336);