#413641

order	512
nilpotency class	3
isoclinism	isoclinic class of #395169
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

860 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)(13,14,18,21)(15,19,25,22)(16,26,20,17)(23,27,28,24);(1,2,4,3)(5,7)(6,9)(8,11)(10,12)(13,14,17,20,18,21,26,16)(15,22,24,23,25,19,27,28);(1,3,4,2)(7,11)(9,12)(13,15,17,24,18,25,26,27)(14,19,20,28,21,22,16,23);(13,16,17,14,18,20,26,21)(15,23,24,19,25,28,27,22)

Polycyclic presentation:

2^9|a^2=g,b^2=e,c^2=e,d^2=h,e^2=i,h^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,h^a=hi,c^b=cg,d^b=di,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, 320837);