#501033

order	512
nilpotency class	4
isoclinism	isoclinic class of #499769
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄×D₈
commutator, Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#3416	C₂
LCS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ #1764 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

981 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)(5,6,9,14,11,15,24,10)(7,16,18,27,19,30,25,12)(8,20,21,29,23,28,22,13)(17,26,32,35,34,36,33,31);(2,4)(6,10)(7,12)(9,24)(13,20)(14,15)(16,25)(17,26)(18,30)(19,27)(21,22)(28,29)(31,32)(33,35)(34,36);(5,7,11,19)(6,12,15,27)(8,17,23,34)(9,25,24,18)(10,16,14,30)(13,26,29,36)(20,31,28,35)(21,33,22,32);(2,4)(5,8)(6,13)(7,17)(9,22)(10,20)(11,23)(12,26)(14,28)(15,29)(16,31)(18,33)(19,34)(21,24)(25,32)(27,36)(30,35)

Polycyclic presentation:

2^9|a^2=f,c^2=i,e^2=i,f^2=i,g^2=hi,h^2=i,b^a=be,c^a=cf,d^a=de,c^b=cg,e^b=ei,f^b=fi,g^b=gh,h^b=hi,e^c=ei,f^c=fi,g^c=gh,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, 408229);