#401068

order	512
nilpotency class	3
isoclinism	isoclinic class of #396101
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2404
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#23445	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1786 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)(7,11,16,23)(8,18,19,33)(9,14)(10,13)(12,20,26,34)(15,29)(17,25,21,28)(22,27)(24,35)(30,36)(31,32);(1,2)(3,4)(5,7)(6,11,13,23)(8,15)(9,21)(10,16)(12,24,26,36)(14,28,27,25)(17,22)(18,32,33,29)(19,31)(20,35)(30,34);(2,4)(9,22)(11,23)(14,27)(17,21)(18,33)(20,34)(24,36)(30,35);(5,8,10,19)(6,12)(7,15,16,31)(9,20,22,34)(11,24)(13,26)(14,18)(17,30,21,35)(23,36)(25,29)(27,33)(28,32)

Polycyclic presentation:

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

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, 308264);