#448248

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, prime radical, cosocle	C₂⁵	C₂⁴
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

1368 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 32 points:

(2,6)(7,8)(10,13)(14,15)(17,18,22,19)(20,23,28,25)(21,24,29,26)(27,31,32,30);(1,2)(3,7)(4,8)(5,6)(9,10)(11,15)(12,13)(14,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3,5,4)(2,7,6,8)(9,11,12,16)(10,14,13,15)(17,20)(18,23)(19,25)(21,27)(22,28)(24,30)(26,31)(29,32);(1,4,5,3)(2,7,6,8)(10,13)(14,15)(17,21)(18,24)(19,26)(20,27)(22,29)(23,30)(25,31)(28,32)

Polycyclic presentation:

2^9|a^2=i,b^2=i,c^2=e,d^2=eg,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de,h^c=hi,f^d=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, 355444);