#413639

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

852 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(2,3)(5,6,8,10)(7,9,11,12)(13,14)(15,22)(16,28)(17,32)(18,34)(19,24)(20,26)(21,31)(23,33)(25,39)(27,38)(29,35)(30,37)(36,41)(40,42)(43,44);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,15,18,26,19,27,33,17)(14,20,23,22,24,32,34,38)(16,25,30,40,31,41,44,29)(21,35,37,36,28,42,43,39);(1,3,4,2)(7,11)(9,12)(13,16,18,30,19,31,33,44)(14,21,23,37,24,28,34,43)(15,25,26,40,27,41,17,29)(20,35,22,36,32,42,38,39);(13,17,33,27,19,26,18,15)(14,22,34,20,24,38,23,32)(16,29,44,41,31,40,30,25)(21,36,43,35,28,39,37,42)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 320835);