#429574

order	512
nilpotency class	3
isoclinism	isoclinic class of #417570
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	#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

1043 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)(7,12)(8,20,22,32)(9,14)(10,15)(11,16)(13,23,30,25)(17,33,21,36)(18,27)(19,28)(24,31)(26,34,29,35);(5,7,10,19,11,9,18,24)(6,12,15,28,16,14,27,31)(8,21,23,35,22,17,25,34)(13,26,20,33,30,29,32,36);(5,8)(6,13)(7,17)(9,21)(10,25)(11,22)(12,26)(14,29)(15,32)(16,30)(18,23)(19,35)(20,27)(24,34)(28,36)(31,33);(2,4)(5,9,10,24,11,7,18,19)(6,14,15,31,16,12,27,28)(8,21,25,34,22,17,23,35)(13,29,32,33,30,26,20,36)

Polycyclic presentation:

2^9|a^2=e,b^2=f,d^2=ef,f^2=i,b^a=be,c^a=cf,d^a=dh,g^a=gi,c^b=cg,d^b=de,g^b=gi,e^c=ei,f^c=fi,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, 336770);