#415117

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	#2495
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

846 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)(5,6,11,17,12,16,10,18)(7,19,25,36,22,13,23,32)(8,14,29,34,21,33,26,31)(9,27,28,15,24,35,20,30);(5,7)(6,13)(8,24)(9,21)(10,25)(11,23)(12,22)(14,27)(15,31)(16,19)(17,36)(18,32)(20,26)(28,29)(30,34)(33,35);(2,4)(5,8)(6,14,16,33)(7,20)(9,23)(10,29)(11,26)(12,21)(13,30,19,15)(17,31,18,34)(22,28)(24,25)(27,32,35,36);(5,9,11,28,12,24,10,20)(6,15,18,27,16,30,17,35)(7,21,23,29,22,8,25,26)(13,31,32,14,19,34,36,33)

Polycyclic presentation:

2^9|a^2=fh,c^2=e,d^2=h,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,h^a=hi,c^b=cg,f^b=fi,g^b=gi,d^c=di,f^c=fi,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, 322313);