#8930337

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
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

4125 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)(3,8)(4,12)(5,7)(6,11)(9,13)(10,14)(15,16)(19,25)(23,30)(26,28)(31,32)(33,34,38,40)(35,39,43,48)(36,41,44,45)(37,46,47,42);(1,3)(2,6)(4,10)(5,11)(7,8)(9,14)(12,15)(13,16)(17,18)(19,23)(20,27)(21,22)(24,29)(25,30)(26,32)(28,31)(33,35,38,43)(34,39,40,48)(36,37,44,47)(41,46,45,42);(2,7)(4,13)(6,8)(10,16)(18,22)(19,25)(24,27)(26,28)(34,40)(35,43)(37,47)(41,45);(1,2)(3,6)(4,9)(5,7)(8,11)(10,14)(12,13)(15,16)(17,19,21,25)(18,23,22,30)(20,28,29,26)(24,32,27,31)(33,34)(35,39)(36,45)(37,42)(38,40)(41,44)(43,48)(46,47);(1,3)(2,8)(4,10)(5,11)(6,7)(9,15)(12,14)(13,16)(17,20)(18,24)(19,26)(21,29)(22,27)(23,31)(25,28)(30,32)(33,36,38,44)(34,41,40,45)(35,37,43,47)(39,46,48,42);(1,4,5,13)(2,9,7,12)(3,10,11,16)(6,14,8,15)(17,18)(19,23)(20,24)(21,22)(25,30)(26,31)(27,29)(28,32)(33,37)(34,42)(35,44)(36,43)(38,47)(39,41)(40,46)(45,48)

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal normal subgroup of	#72219279, #72219281, #72219283
maximal quotient of	#72219273

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 8837533);