#6369934

order	512
nilpotency class	3
isoclinism	isoclinic class of #576
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₈	C₂²×D₄
commutator	C₄	C₂³×C₄²
Frattini, cosocle	C₂²×C₄	C₂⁵
socle	C₂²	C₂²×C₄×D₄
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₈ ◁ C₂²×#131 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

3024 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=fg,f^2=i,b^a=bf,f^b=fi,g^b=gi,e^d=ei

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72226779, #72226792, #72226793
maximal normal subgroup of	#72226779, #72226792, #72226793

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6277130);