#8092789

order	512
nilpotency class	2
isoclinism	isoclinic class of #52308
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁵
commutator, Frattini, socle, cosocle	C₂³	C₂⁶
prime radical	#52561	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4069 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)(3,7)(4,8)(5,6)(9,12)(10,14)(11,13)(15,16)(17,18)(19,24)(20,28)(21,25)(22,30)(23,26)(27,31)(29,32)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(2,6)(7,12)(8,13)(14,16)(17,19,23,20)(18,24,26,28)(21,29,22,27)(25,32,30,31)(33,35,38,37)(34,39,42,41)(36,44,45,43)(40,47,48,46);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16)(17,20,23,19)(18,24,26,28)(21,27,22,29)(25,32,30,31)(33,36,38,45)(34,40,42,48)(35,43,37,44)(39,46,41,47);(2,6)(3,9)(8,13)(10,15)(17,21,23,22)(18,25,26,30)(19,27,20,29)(24,31,28,32)(33,37,38,35)(34,41,42,39)(36,43,45,44)(40,46,48,47);(2,6)(7,12)(8,13)(14,16)(17,22,23,21)(18,25,26,30)(19,27,20,29)(24,32,28,31)(34,42)(39,41)(40,48)(46,47);(1,4,5,11)(2,8,6,13)(3,10,9,15)(7,14,12,16)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 7999985);