#428077

order	512
nilpotency class	3
isoclinism	isoclinic class of #428071
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2504
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#535	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1179 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:

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

Polycyclic presentation:

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

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, 335273);