#420053

order	512
nilpotency class	3
isoclinism	isoclinic class of #418466
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1450 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:

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

Polycyclic presentation:

2^9|a^2=ef,d^2=e,e^2=i,b^a=be,c^a=cf,d^a=dh,h^a=hi,c^b=cg,d^b=de,e^b=ei,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, 327249);