#415468

order	512
nilpotency class	3
isoclinism	isoclinic class of #396117
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

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

900 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:

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

Polycyclic presentation:

2^9|a^2=gh,b^2=e,d^2=h,e^2=i,f^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 322664);