#451752

order	512
nilpotency class	3
isoclinism	isoclinic class of #428059
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2749
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

889 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 358948);