#263743

order	512
nilpotency class	3
isoclinism	isoclinic class of #263739
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2408
commutator, Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	#447	D₄
LCS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

781 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=i,b^2=e,d^2=eh,e^2=h,f^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=eh,f^a=fi,g^a=gh,c^b=cg,f^b=fh,g^b=gi,f^c=fi,g^c=gi

Conjugacy classes & character table

59 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 170939);