#418307

order	512
nilpotency class	3
isoclinism	isoclinic class of #417489
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
prime radical	C₂×#2010	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1754 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)(16,30)(24,35)(29,36)(31,34);(1,2)(3,4)(5,6,10,14)(7,15,18,28)(8,12,19,26)(9,21,23,13)(11,22,25,17)(16,29,30,36)(20,33,32,27)(24,34,35,31);(5,7)(6,11)(8,16)(9,22)(10,18)(12,24)(13,28)(14,25)(15,21)(17,23)(19,30)(20,34)(26,35)(27,36)(29,33)(31,32);(1,3,4,2)(5,8)(6,12)(7,16)(9,20)(10,19)(11,24)(13,27)(14,26)(15,29)(17,31)(18,30)(21,33)(22,34)(23,32)(25,35)(28,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72313084, #72313145, #72313170
maximal normal subgroup of	#72313084, #72313145, #72313170

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325503);