#8455735

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	#2671	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3957 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)(4,9)(5,6)(7,10)(8,15)(12,14)(13,16)(17,18,22,26)(19,27,29,23)(20,24,30,32)(21,25,31,28)(33,34,38,37)(35,41,42,39)(36,40,44,45)(43,47,48,46);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,19,22,29)(18,23,26,27)(20,28,30,25)(21,24,31,32)(35,42)(36,44)(39,41)(40,45);(1,4,7,5)(2,9,10,6)(3,12,15,13)(8,16,11,14)(17,18)(19,23)(20,24)(21,25)(22,26)(27,29)(28,31)(30,32)(33,35)(34,39)(36,43)(37,41)(38,42)(40,46)(44,48)(45,47);(1,5,7,4)(2,9,10,6)(3,13,15,12)(8,16,11,14)(17,18)(19,27)(20,24)(21,28)(22,26)(23,29)(25,31)(30,32)(33,36,38,44)(34,40,37,45)(35,43,42,48)(39,46,41,47);(1,3)(2,8)(4,12)(5,13)(6,14)(7,15)(9,16)(10,11)(17,20,22,30)(18,24,26,32)(19,28,29,25)(21,23,31,27);(1,6,7,9)(2,5,10,4)(3,14,15,16)(8,13,11,12)(17,21)(18,25)(19,24)(20,23)(22,31)(26,28)(27,30)(29,32)(33,37,38,34)(35,41,42,39)(36,40,44,45)(43,46,48,47)

Polycyclic presentation:

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

Conjugacy classes & character table

89 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8362931);