#419137

order	512
nilpotency class	3
isoclinism	isoclinic class of #417589
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂⁴	C₂²×D₄
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2407	C₂
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×C₄×D₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2272 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=ef,b^2=e,d^2=i,f^2=i,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=dei,f^c=fi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 326333);