#439373

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:

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

Polycyclic presentation:

2^9|a^2=efi,b^2=efg,c^2=efi,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, 346569);