#413640

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

896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=g,b^2=ei,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, 320836);