#423847

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	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1300 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=i,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, 331043);