#8762056

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	#44153	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4186 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=g,c^2=gh,d^2=g,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^b=fgi,d^c=dgi,e^c=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, 8669252);