#428406

order	512
nilpotency class	3
isoclinism	isoclinic class of #421163
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1033 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:

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

Polycyclic presentation:

2^9|a^2=ef,c^2=i,d^2=f,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,c^b=cg,d^b=de,e^b=ei,f^b=fi,g^b=gi,h^b=hi,e^c=ei,g^c=gi,g^d=gi,h^d=hi

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 335602);