#8093594

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	#2619	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3913 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8000790);