#424736

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2535
prime radical	C₂×#525	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1116 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:

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

Polycyclic presentation:

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

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, 331932);