#8430415

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	#2410	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4242 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 6 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=g,c^2=gh,d^2=h,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, 8337611);