#414857

order	512
nilpotency class	3
isoclinism	isoclinic class of #394335
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#121
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1080 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 44
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 4 generators on 44 points:

(1,2)(3,4)(5,6,8,10)(7,11,12,9)(13,14,18,21)(15,23,26,38)(16,27,30,20)(17,31,33,22)(19,32,35,25)(24,40,28,39)(29,37,44,42)(34,41,36,43);(13,15,18,26)(14,19,21,35)(16,28,30,24)(17,32,33,25)(20,36,27,34)(22,38,31,23)(29,41,44,43)(37,39,42,40);(2,4)(13,16,17,29,18,30,33,44)(14,20,22,37,21,27,31,42)(15,24,25,41,26,28,32,43)(19,34,23,39,35,36,38,40);(5,7,8,12)(6,9,10,11)(14,21)(19,35)(20,27)(22,31)(23,38)(34,36)(37,42)(39,40)

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=e,d^2=hi,e^2=i,b^a=be,c^a=cf,d^a=dh,e^a=ei,c^b=cg,e^b=ei,f^c=fi,g^c=gi

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, 322053);