#8458629

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	#59394	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4099 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)(3,12)(4,8)(5,9)(6,10)(7,11)(13,14)(15,16)(17,18)(19,27)(20,25)(21,22)(23,26)(24,31)(28,30)(29,32)(36,44)(40,47)(43,48)(45,46);(3,4)(8,12)(13,15)(14,16)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,34)(35,39)(36,40)(37,41)(38,42)(43,46)(44,47)(45,48);(1,3,7,4)(2,8,11,12)(5,13,10,15)(6,14,9,16)(17,20,23,21)(18,25,26,22)(19,28,31,29)(24,32,27,30)(33,35)(34,37)(36,43)(38,41)(39,42)(40,45)(44,48)(46,47);(1,4,7,3)(2,8,11,12)(5,15,10,13)(6,14,9,16)(17,21,23,20)(18,25,26,22)(19,29,31,28)(24,32,27,30)(33,35)(34,39)(36,43)(37,42)(38,41)(40,46)(44,48)(45,47);(1,5,7,10)(2,9,11,6)(3,13,4,15)(8,16,12,14)(17,19)(18,24)(20,28)(21,29)(22,30)(23,31)(25,32)(26,27)(33,36,38,44)(34,40,42,47)(35,43,41,48)(37,45,39,46);(1,6,7,9)(2,10,11,5)(3,14,4,16)(8,15,12,13)(17,22,23,25)(18,21,26,20)(19,30,31,32)(24,29,27,28)(33,37,38,39)(34,41,42,35)(36,45,44,46)(40,48,47,43)

Polycyclic presentation:

2^9|c^2=gh,d^2=gh,e^2=i,f^2=gi,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, 8365825);