#418468

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1729 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72314443, #72314463, #72314468
maximal normal subgroup of	#72314443, #72314463, #72314468
maximal quotient of	#72314520

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325664);