#118371

order	512
nilpotency class	3
isoclinism	isoclinic class of #118368
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1439
commutator	C₂³×C₄	C₂²×C₄
Frattini, cosocle	C₂⁴×C₄	C₂³
prime radical	C₂⁵	#41
LCS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴×C₄ ◁ G
LECS	G ▷ C₂⁴×C₄ ▷ C₂² ▷ C₁

696 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2)(3,6)(4,11,13,7)(5,8)(9,16,12,14)(10,15)(17,18,22,26,23,27,21,37)(19,28,31,39,32,43,30,24)(20,33,35,25,36,40,34,41)(29,38,45,42,46,48,44,47);(1,3)(2,6)(4,12,13,9)(5,10)(7,14,11,16)(8,15)(17,19,21,30,23,32,22,31)(18,24,26,28,27,39,37,43)(20,29,34,44,36,46,35,45)(25,38,40,42,41,48,33,47);(1,4)(2,7)(3,9)(5,13)(6,14)(8,11)(10,12)(15,16)(17,20,23,36)(18,25,27,41)(19,29,32,46)(21,34,22,35)(24,38,39,48)(26,40,37,33)(28,42,43,47)(30,44,31,45)

Polycyclic presentation:

2^9|a^2=g,b^2=d,c^2=h,d^2=h,e^2=h,g^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=ehi,f^a=fi,c^b=cf,f^b=fi,d^c=di,e^c=eh,g^c=gi

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 25567);