#473596

order	512
nilpotency class	3
isoclinism	isoclinic class of #455641
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂²×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2826
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	2₋¹⁺⁴
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

679 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=efi,d^2=ef,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,f^a=fi,c^b=cg,e^b=ei,f^b=fi,h^b=hi,d^c=de,e^c=ei,f^c=fi,h^c=hi,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, 380792);