#413466

order	512
nilpotency class	3
isoclinism	isoclinic class of #395174
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

974 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|b^2=ef,c^2=ef,d^2=h,e^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,d^b=di,e^c=ei,f^c=fi

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, 320662);