#424337

order	512
nilpotency class	3
isoclinism	isoclinic class of #422540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2669
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23406	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1505 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:

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

Polycyclic presentation:

2^9|a^2=e,c^2=ef,d^2=e,e^2=i,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,e^b=ei,f^b=fi,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, 331533);