#466060

order	512
nilpotency class	3
isoclinism	isoclinic class of #455828
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄²

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

1030 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:

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

Polycyclic presentation:

2^9|a^2=i,b^2=fg,c^2=e,d^2=f,e^2=i,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,h^b=hi,d^c=de,h^c=hi,e^d=ei

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, 373256);