#423851

order	512
nilpotency class	3
isoclinism	isoclinic class of #418976
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3041	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1395 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)(3,4)(5,6,9,15,11,16,10,12)(7,17,20,29,18,26,21,13)(8,22,23,27,25,14,24,30)(19,33,35,28,32,31,34,36);(6,12)(7,18)(8,23)(9,10)(13,26)(15,16)(17,29)(19,34)(24,25)(27,30)(31,33)(32,35);(5,7,10,21,11,18,9,20)(6,13,15,17,16,29,12,26)(8,19,24,34,25,32,23,35)(14,28,30,31,22,36,27,33);(2,4)(5,8,10,24,11,25,9,23)(6,14,12,27,16,22,15,30)(7,19,21,34,18,32,20,35)(13,28,26,33,29,36,17,31)

Polycyclic presentation:

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

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, 331047);