#9481528

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	#53894	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3767 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(17,18)(19,20)(21,24)(22,25)(23,27)(26,28)(29,31)(30,32)(33,34,39,42)(35,37,41,38)(36,44,47,40)(43,46,48,45);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28)(33,35,39,41)(34,38,42,37)(36,43,47,48)(40,46,44,45);(2,7)(4,13)(6,8)(10,16)(21,29)(24,31)(26,32)(28,30)(33,35)(34,37)(36,43)(38,42)(39,41)(40,45)(44,46)(47,48);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,18)(19,23)(20,27)(21,24)(22,25)(26,30)(28,32)(29,31)(33,36,39,47)(34,40,42,44)(35,43,41,48)(37,45,38,46);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,20,22,23)(18,19,25,27)(21,28,29,30)(24,26,31,32)(33,37,39,38)(34,41,42,35)(36,45,47,46)(40,48,44,43);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)(33,38,39,37)(34,35,42,41)(36,46,47,45)(40,43,44,48)

Polycyclic presentation:

2^9|a^2=i,b^2=gh,d^2=i,e^2=gh,f^2=g,b^a=bg,c^a=ch,d^a=di,f^a=fh,c^b=ci,e^b=eghi,d^c=dh,f^c=fgi,e^d=egi

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 9388724);