#8105630

order	512
nilpotency class	2
isoclinism	isoclinic class of #53390
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

3994 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)(3,11)(4,5)(6,10)(7,9)(8,14)(12,13)(15,16)(17,18,23,26)(19,22,25,21)(20,28,30,24)(27,31,32,29)(33,34,38,42)(35,39,40,36)(37,41,47,48)(43,46,45,44);(1,3,7,14)(2,8,9,11)(4,12,10,16)(5,15,6,13)(17,19,23,25)(18,21,26,22)(20,27,30,32)(24,31,28,29)(33,35)(34,39)(36,42)(37,45)(38,40)(41,44)(43,47)(46,48);(1,4,7,10)(2,5,9,6)(3,12,14,16)(8,15,11,13)(17,20)(18,24)(19,27)(21,31)(22,29)(23,30)(25,32)(26,28)(33,36,38,39)(34,40,42,35)(37,46,47,44)(41,43,48,45);(2,9)(4,10)(8,11)(12,16)(17,21,23,22)(18,25,26,19)(20,29,30,31)(24,27,28,32)(33,37,38,47)(34,41,42,48)(35,43,40,45)(36,44,39,46);(1,5,7,6)(2,4,9,10)(3,13,14,15)(8,16,11,12)(17,22,23,21)(18,19,26,25)(20,29,30,31)(24,27,28,32)(34,42)(35,40)(41,48)(43,45);(1,6,7,5)(2,10,9,4)(3,13,14,15)(8,16,11,12)

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,e^2=g,f^2=ghi,b^a=bg,c^a=ch,d^a=dg,e^a=eh,c^b=ci,d^b=dh,e^b=egi,d^c=dgi

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, 8012826);