#414979

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×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1248 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)(7,18,23,13)(8,24,20,35)(9,28)(10,17)(11,30)(12,16)(14,26,31,27)(15,19)(21,33,22,32)(25,36)(29,34);(2,4)(5,7)(6,13)(8,25)(9,20)(10,22)(11,21)(12,23)(14,34)(15,31)(16,18)(17,33)(19,27)(24,28)(26,29)(30,32)(35,36);(5,8)(6,14)(7,19)(9,22)(10,27)(11,26)(12,20)(13,28)(15,32)(16,31)(17,35)(18,36)(21,25)(23,29)(24,30)(33,34);(5,9,11,19,12,25,10,29)(6,15,17,28,16,34,30,36)(7,20,21,27,23,8,22,26)(13,31,33,24,18,14,32,35)

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,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, 322175);