#419911

order	512
nilpotency class	3
isoclinism	isoclinic class of #418492
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	#23407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1598 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:

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

Polycyclic presentation:

2^9|a^2=f,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,d^b=de,e^b=ei,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, 327107);