#428854

order	512
nilpotency class	3
isoclinism	isoclinic class of #417541
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	#535
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	#2598
prime radical	C₂×#575	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

967 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)(5,6,9,14,11,15,10,23)(7,16,20,26,19,30,17,12)(8,13,22,28,21,27,24,29)(18,31,34,35,33,36,32,25);(7,17,19,20)(8,21)(12,16,26,30)(13,27)(18,32,33,34)(22,24)(25,31,35,36)(28,29);(5,7)(6,12)(8,18)(9,17)(10,20)(11,19)(13,25)(14,30)(15,26)(16,23)(21,33)(22,32)(24,34)(27,35)(28,36)(29,31);(2,4)(5,8,9,22,11,21,10,24)(6,13,14,28,15,27,23,29)(7,18,17,32,19,33,20,34)(12,25,30,36,26,35,16,31)

Polycyclic presentation:

2^9|a^2=f,b^2=e,d^2=ef,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=dei,e^c=ei,f^c=fi,g^c=gi,g^d=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, 336050);