#428075

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:

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

Polycyclic presentation:

2^9|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, 335271);