#397749

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1212 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=f,c^2=ei,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,h^a=hi,c^b=cg,g^b=gi,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, 304945);