#420454

order	512
nilpotency class	3
isoclinism	isoclinic class of #420453
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	#60456	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1710 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=e,d^2=ei,f^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,h^d=hi

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, 327650);