#495538

order	512
nilpotency class	4
isoclinism	isoclinic class of #33691
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#3109
commutator	C₂×C₈	C₂³×C₄
Frattini, cosocle	C₂²×C₈	C₂⁴
prime radical	C₂×#469	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂²×C₈ ▷ C₄ ▷ C₂ ▷ C₁

831 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 402734);