#376199

order	512
nilpotency class	3
isoclinism	isoclinic class of #28182
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1577 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 283395);