#424744

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1068 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

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

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, 331940);