#8893631

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	#58027	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

3927 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)(4,15)(5,10)(6,7)(8,14)(9,13)(12,16)(17,18,22,25)(19,23,26,30)(20,21,24,29)(27,28,31,32)(33,34,38,36)(35,41,43,39)(37,40,45,44)(42,47,48,46);(1,3,7,14)(2,8,6,11)(4,12,13,5)(9,16,15,10)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(37,45)(40,44)(42,48)(46,47);(2,6)(8,11)(9,15)(10,16)(19,26)(23,30)(27,31)(28,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48);(1,4)(2,9)(3,12)(5,14)(6,15)(7,13)(8,16)(10,11)(17,20)(18,21)(19,27)(22,24)(23,28)(25,29)(26,31)(30,32)(33,36,38,34)(35,41,43,39)(37,44,45,40)(42,47,48,46);(1,5,7,12)(2,10,6,16)(3,13,14,4)(8,15,11,9)(17,21)(18,24)(19,28)(20,25)(22,29)(23,31)(26,32)(27,30)(33,37)(34,40)(35,42)(36,44)(38,45)(39,46)(41,47)(43,48);(1,6,7,2)(3,11,14,8)(4,15,13,9)(5,16,12,10)(17,19)(18,23)(20,27)(21,28)(22,26)(24,31)(25,30)(29,32)(33,35)(34,39)(36,41)(37,42)(38,43)(40,46)(44,47)(45,48)

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,d^2=gh,e^2=g,f^2=g,b^a=bg,c^a=ch,d^a=dg,f^a=fh,c^b=ci,e^b=eh,f^c=fgi,e^d=egi

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8800827);