#450591

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

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

855 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)(5,6,12,29,11,17,10,16)(7,18,28,13,22,34,21,31)(8,23,27,15,24,14,9,25)(19,33,26,30,20,36,35,32);(5,7,12,21,11,22,10,28)(6,13,29,18,17,31,16,34)(8,19,27,35,24,20,9,26)(14,32,25,36,23,30,15,33);(5,8,12,27,11,24,10,9)(6,14,29,25,17,23,16,15)(7,19,21,35,22,20,28,26)(13,30,18,33,31,32,34,36);(2,4)(5,9,10,24,11,27,12,8)(6,15,16,23,17,25,29,14)(7,20,21,26,22,19,28,35)(13,30,18,33,31,32,34,36)

Polycyclic presentation:

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

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, 357787);