#455006

order	512
nilpotency class	3
isoclinism	isoclinic class of #454057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35150	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1450 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 362202);