#397514

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×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1262 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=ef,d^2=e,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,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, 304710);