#418466

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
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1801 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)(7,13)(8,20)(9,12)(10,15)(11,16)(14,23)(17,31)(18,25)(19,28)(21,30)(22,29)(24,27)(26,32)(33,34)(35,36);(1,2)(3,4)(6,12)(8,21)(9,10)(13,25)(15,16)(17,32)(18,24)(20,29)(22,23)(27,28)(31,34)(33,35);(2,4)(5,7)(6,13)(8,17)(9,18)(10,24)(11,19)(12,25)(14,26)(15,27)(16,28)(20,34)(21,32)(22,33)(23,35)(29,31)(30,36);(5,8,9,21,11,22,10,23)(6,14,15,29,16,30,12,20)(7,17,18,32,19,33,24,35)(13,26,27,31,28,36,25,34)

Polycyclic presentation:

2^9|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	#72256053, #72262411, #72262422, …
maximal normal subgroup of	#72256053, #72262411, #72262422, …
maximal quotient of	#72314562

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 325662);