#418315

order	512
nilpotency class	3
isoclinism	isoclinic class of #417597
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2403
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1782 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313074, #72313145, #72313151, …
maximal normal subgroup of	#72313074, #72313145, #72313151, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325511);