#439372

order	512
nilpotency class	3
isoclinism	isoclinic class of #419919
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1164 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:

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

Polycyclic presentation:

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

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, 346568);