#426458

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

796 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)(5,6,10,15,11,16,18,28)(7,17,21,31,22,34,27,12)(8,23,9,14,24,13,26,32)(19,35,36,30,20,33,25,29);(5,7,18,21,11,22,10,27)(6,12,28,17,16,31,15,34)(8,19,26,36,24,20,9,25)(13,30,14,33,23,29,32,35);(5,8,18,26,11,24,10,9)(6,13,28,14,16,23,15,32)(7,19,21,36,22,20,27,25)(12,29,17,35,31,30,34,33);(2,4)(5,9,10,24,11,26,18,8)(6,14,15,13,16,32,28,23)(7,20,21,25,22,19,27,36)(12,30,17,33,31,29,34,35)

Polycyclic presentation:

2^9|a^2=e,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, 333654);