#419891

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	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1623 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)(7,16)(8,20,23,30)(9,14)(10,15)(11,18)(12,22,31,32)(13,25)(17,26,33,34)(19,27)(21,35,36,29)(24,28);(1,2)(3,4)(5,7,10,19)(6,11,15,28)(8,21,23,36)(9,24,25,18)(12,29,31,35)(13,27,14,16)(17,32,33,22)(20,34,30,26);(2,4)(5,6)(7,11)(8,12)(9,13)(10,15)(14,25)(16,18)(17,26)(19,28)(20,32)(21,29)(22,30)(23,31)(24,27)(33,34)(35,36);(5,8,9,22,10,23,25,32)(6,12,13,30,15,31,14,20)(7,17,18,21,19,33,24,36)(11,26,16,29,28,34,27,35)

Polycyclic presentation:

2^9|a^2=f,b^2=i,d^2=e,e^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,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, 327087);