#413094

order	512
nilpotency class	3
isoclinism	isoclinic class of #397514
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2428
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#3155	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1576 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:

(2,3)(5,6,10,16,11,15,12,7)(8,17,20,13,22,29,21,28)(9,23,24,30,25,18,26,14)(19,27,32,35,34,36,33,31);(1,2,4,3)(5,7,10,15,11,16,12,6)(8,17,20,28,22,29,21,13)(9,18,24,30,25,23,26,14)(19,31,32,36,34,35,33,27);(1,3,4,2)(5,8)(6,13,15,28)(7,17,16,29)(9,19)(10,20)(11,22)(12,21)(14,27,30,36)(18,31,23,35)(24,32)(25,34)(26,33);(5,9,10,24,11,25,12,26)(6,14,7,18,15,30,16,23)(8,19,20,32,22,34,21,33)(13,27,17,31,28,36,29,35)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320290);