#420464

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	#60454	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1765 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=ei,d^2=ei,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.

Relationships

maximal subgroup of	#72316858
maximal normal subgroup of	#72316858
maximal quotient of	#72316862, #72316977, #72317020

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 327660);