#406105

order	512
nilpotency class	3
isoclinism	isoclinic class of #400057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2631
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#41	C₂³
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

979 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)(7,13)(8,21,23,26)(9,12)(10,15)(11,16)(14,24,30,22)(17,28)(18,32,34,35)(19,25)(20,29)(27,33,36,31);(1,2)(3,4)(6,12,16,15)(7,17,20,19)(14,26,30,21)(18,31,34,33);(2,4)(5,7,11,20)(6,13,16,29)(8,18,23,34)(9,19,10,17)(12,25,15,28)(14,27,30,36)(21,35,26,32)(22,33,24,31);(5,8,9,22,11,23,10,24)(6,14,15,21,16,30,12,26)(7,18,19,33,20,34,17,31)(13,27,28,35,29,36,25,32)

Polycyclic presentation:

2^9|a^2=f,b^2=eh,c^2=i,d^2=e,e^2=i,g^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,g^c=gi,h^c=hi,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, 313301);