#417509

order	512
nilpotency class	3
isoclinism	isoclinic class of #417508
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂⁵	C₂⁴
prime radical	C₂²×B(4,𝔽₂)	C₂
LCS	G ▷ C₂⁵ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁵ ▷ C₂ ▷ C₁

3248 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

2^9|b^2=i,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, 324705);