#420463

order	512
nilpotency class	3
isoclinism	isoclinic class of #420461
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	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1751 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)(8,13)(9,22)(11,16,24,27)(14,18)(17,28,19,26)(20,30)(21,31)(23,32)(25,33)(29,35,36,34);(1,2)(3,4)(5,7)(6,12)(8,17)(9,10)(11,23)(13,28)(14,27)(15,22)(16,32)(18,24)(19,20)(21,35)(25,34)(26,30)(29,33)(31,36);(2,4)(5,8)(6,13,15,30)(7,17)(9,19)(10,20)(11,25)(12,26,22,28)(14,29,32,36)(16,33,27,31)(18,35)(21,24)(23,34);(5,9,10,7)(6,14)(8,19,20,17)(11,23,24,18)(12,27)(13,29)(15,32)(16,22)(21,35,25,34)(26,31)(28,33)(30,36)

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327659);