#454654

order	512
nilpotency class	3
isoclinism	isoclinic class of #454057
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	#35149	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2027 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)(13,14,18,22,19,16,15,20)(17,21,23,27,25,24,26,28);(1,2)(3,4)(6,9)(10,12)(13,15,19,18)(14,20,16,22)(17,23,25,26)(21,27,24,28);(2,4)(5,7)(6,10)(8,11)(9,12)(13,16)(14,19)(15,20)(17,24)(18,22)(21,25)(23,27)(26,28);(7,11)(10,12)(13,17,19,25)(14,21,16,24)(15,23,18,26)(20,27,22,28)

Polycyclic presentation:

2^9|a^2=ef,b^2=i,d^2=i,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, 361850);