#427991

order	512
nilpotency class	3
isoclinism	isoclinic class of #419800
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2535
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

1259 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)(5,6,10,15,12,17,11,16)(7,18,21,33,23,35,22,13)(8,24,26,14,28,30,27,9)(19,31,36,34,32,20,25,29);(5,7)(6,13)(8,25)(9,29)(10,22)(11,21)(12,23)(14,34)(15,35)(16,18)(17,33)(19,27)(20,24)(26,32)(28,36)(30,31);(5,8,12,28)(6,14,17,9)(7,19,23,32)(10,27,11,26)(13,31,33,20)(15,24,16,30)(18,29,35,34)(21,25,22,36);(2,4)(5,9)(6,8)(7,20,23,31)(10,24)(11,30)(12,14)(13,32,33,19)(15,26)(16,27)(17,28)(18,25,35,36)(21,29,22,34)

Polycyclic presentation:

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

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, 335187);