#412606

order	512
nilpotency class	3
isoclinism	isoclinic class of #395166
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

986 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72321698, #72321734, #72321742, …
maximal normal subgroup of	#72321698, #72321734, #72321742, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 319802);