#8425623

order	512
nilpotency class	2
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	C₂²×D₄	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3883 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=h,c^2=hi,d^2=g,e^2=h,f^2=g,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,f^d=fgi

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8332819);