#414980

order	512
nilpotency class	3
isoclinism	isoclinic class of #396118
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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1238 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)(7,19,23,13)(8,24,21,36)(9,28)(10,17)(11,18)(12,16)(14,27,32,30)(15,29)(20,35)(22,34,26,33)(25,31);(2,4)(5,7)(6,13)(8,25)(9,21)(10,26)(11,22)(12,23)(14,35)(15,32)(16,19)(17,34)(18,33)(20,27)(24,28)(29,30)(31,36);(5,8)(6,14)(7,20)(9,22)(10,30)(11,27)(12,21)(13,31)(15,34)(16,32)(17,36)(18,24)(19,28)(23,29)(25,26)(33,35);(5,9,11,29,12,25,10,20)(6,15,17,28,16,35,18,31)(7,21,22,30,23,8,26,27)(13,32,34,24,19,14,33,36)

Polycyclic presentation:

2^9|a^2=efh,d^2=h,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,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, 322176);