#423846

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×#3045	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1356 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)(5,6,9,14,11,15,10,27)(7,16,19,29,21,17,20,12)(8,22,24,36,26,13,25,23)(18,31,32,28,34,30,33,35);(5,6)(7,17)(8,23)(9,27)(10,14)(11,15)(12,20)(13,24)(16,21)(18,28)(19,29)(22,25)(26,36)(30,33)(31,32)(34,35);(5,7,10,20,11,21,9,19)(6,12,14,16,15,29,27,17)(8,18,25,33,26,34,24,32)(13,28,23,30,22,35,36,31);(2,4)(5,8,9,24,11,26,10,25)(6,13,14,23,15,22,27,36)(7,18,19,32,21,34,20,33)(12,28,16,30,29,35,17,31)

Polycyclic presentation:

2^9|a^2=e,c^2=f,d^2=e,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,d^b=de,e^b=ei,g^b=gi,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, 331042);