#8092703

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	C₂×#121	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4020 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=i,c^2=g,d^2=g,f^2=h,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, 7999899);