#426017

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2482
prime radical	#3449	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1018 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)(7,11)(8,18,19,28)(9,13)(10,14)(12,20,27,21)(15,30,31,35)(16,24)(17,25)(22,29)(23,32,26,33)(34,36);(5,7,9,16,10,17,22,34)(6,11,13,24,14,25,29,36)(8,15,21,33,19,31,20,32)(12,26,28,30,27,23,18,35);(5,8,9,21,10,19,22,20)(6,12,13,28,14,27,29,18)(7,15,16,33,17,31,34,32)(11,23,24,35,25,26,36,30);(2,4)(5,6)(7,11)(8,12)(9,13)(10,14)(15,23)(16,24)(17,25)(18,20)(19,27)(21,28)(22,29)(26,31)(30,32)(33,35)(34,36)

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 333213);