#473595

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	C₂³×D₄	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

711 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=egh,c^2=ef,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, 380791);