#424261

order	512
nilpotency class	3
isoclinism	isoclinic class of #418464
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23400	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1524 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=efi,d^2=ei,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,g^a=gi,h^a=hi,c^b=cg,d^b=de,f^b=fi,g^c=gi,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, 331457);