#411972

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2563
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1124 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319168);