#397116

order	512
nilpotency class	3
isoclinism	isoclinic class of #396836
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

953 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=ef,c^2=e,d^2=e,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,g^b=gi,f^c=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, 304312);