#459147

order	512
nilpotency class	3
isoclinism	isoclinic class of #455826
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

928 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)(5,6,10,15,12,16,11,9)(7,17,19,29,20,31,27,13)(8,21,23,14,25,26,24,30)(18,28,34,36,22,33,35,32);(5,7,11,19,12,20,10,27)(6,13,9,17,16,29,15,31)(8,22,24,35,25,18,23,34)(14,28,21,36,30,33,26,32);(5,8,12,25)(6,14,16,30)(7,18,20,22)(9,26,15,21)(10,24,11,23)(13,28,29,33)(17,32,31,36)(19,35,27,34);(2,4)(5,9,11,16,12,15,10,6)(7,17,19,29,20,31,27,13)(8,14,24,21,25,30,23,26)(18,33,34,32,22,28,35,36)

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 366343);