#426698

order	512
nilpotency class	3
isoclinism	isoclinic class of #417540
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2487
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

840 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)(5,6,10,15,12,16,11,28)(7,9,18,26,19,27,14,29)(8,20,24,32,23,33,25,13)(17,30,31,35,34,36,21,22);(5,7,10,18,12,19,11,14)(6,9,15,26,16,27,28,29)(8,21,24,17,23,31,25,34)(13,30,20,35,32,36,33,22);(5,8,10,25,12,23,11,24)(6,13,15,33,16,32,28,20)(7,17,18,21,19,34,14,31)(9,22,26,36,27,35,29,30);(2,4)(5,9)(6,14)(7,15)(8,22)(10,26)(11,29)(12,27)(13,31)(16,18)(17,33)(19,28)(20,34)(21,32)(23,35)(24,30)(25,36)

Polycyclic presentation:

2^9|a^2=f,b^2=f,c^2=ef,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^c=ei,f^c=fi,g^c=gi,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, 333894);