#456895

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

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2018	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1203 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:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=ef,c^2=e,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,f^a=fi,g^a=gi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,f^c=fi,e^d=ei,f^d=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, 364091);