#461150

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
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

849 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)(5,6,10,15,12,17,11,16)(7,18,21,32,23,31,22,13)(8,24,27,9,28,29,26,14)(19,34,36,33,35,30,25,20);(5,7,10,22,12,23,11,21)(6,13,15,31,17,32,16,18)(8,25,27,35,28,36,26,19)(9,20,29,30,14,33,24,34);(5,8,10,26,12,28,11,27)(6,14,15,29,17,9,16,24)(7,19,21,25,23,35,22,36)(13,30,18,33,32,34,31,20);(2,4)(5,9,12,14)(6,8,17,28)(7,20,23,33)(10,29,11,24)(13,25,32,36)(15,27,16,26)(18,19,31,35)(21,34,22,30)

Polycyclic presentation:

2^9|a^2=efg,b^2=f,c^2=e,d^2=i,e^2=i,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,e^b=ei,h^b=hi,d^c=de,f^c=fi,e^d=ei,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, 368346);