#473133

order	512
nilpotency class	3
isoclinism	isoclinic class of #456670
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
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

873 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:

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

Polycyclic presentation:

2^9|a^2=fi,b^2=egh,c^2=i,d^2=ef,f^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,g^b=gi,h^b=hi,d^c=de,f^c=fi,h^c=hi,e^d=ei,f^d=fi,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, 380329);