#416418

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#555
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2420
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

920 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)(7,13)(8,22,24,14)(9,26)(10,18)(11,17)(12,16)(15,21)(19,31,34,28)(20,29)(23,30,35,36)(25,32,27,33);(2,4)(5,7,11,9,12,21,10,20)(6,13,17,26,16,15,18,29)(8,23,27,19,24,35,25,34)(14,30,32,31,22,36,33,28);(2,4)(5,8)(6,14)(7,19)(9,23)(10,27)(11,25)(12,24)(13,28)(15,31)(16,22)(17,33)(18,32)(20,35)(21,34)(26,36)(29,30);(5,9,11,21,12,20,10,7)(6,15,18,26,16,13,17,29)(8,23,25,34,24,35,27,19)(14,31,32,36,22,28,33,30)

Polycyclic presentation:

2^9|a^2=e,b^2=efh,d^2=h,g^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=di,e^c=ei,g^c=gi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 323614);