#472246

order	512
nilpotency class	3
isoclinism	isoclinic class of #454681
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2622
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

828 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:

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

Polycyclic presentation:

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

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, 379442);