#427987

order	512
nilpotency class	3
isoclinism	isoclinic class of #419794
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×#2008	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1329 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)(5,6,9,14,10,15,24,12)(7,16,18,27,20,25,19,11)(8,13,22,30,23,31,35,28)(17,29,21,34,33,36,32,26);(5,7)(6,11)(8,21)(9,19)(10,20)(12,16)(13,29)(14,25)(15,27)(17,22)(18,24)(23,32)(26,30)(28,34)(31,36)(33,35);(6,12)(9,24)(11,25)(13,28)(14,15)(16,27)(18,19)(21,32)(22,35)(26,36)(29,34)(30,31);(2,4)(5,8,10,23)(6,13,15,31)(7,17)(9,22,24,35)(11,26)(12,28,14,30)(16,29)(18,21)(19,32)(20,33)(25,36)(27,34)

Polycyclic presentation:

2^9|a^2=e,d^2=ef,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,e^b=ei,e^c=ei,f^c=fi,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, 335183);