#454469

order	512
nilpotency class	3
isoclinism	isoclinic class of #454045
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	C₂⁵	C₂⁴
prime radical	#34984	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1962 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:

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

Polycyclic presentation:

2^9|a^2=e,b^2=ei,d^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^c=de,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, 361665);