#5001735

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, prime radical, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

1362 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 512

A minimal faithful permutation representation — 5 generators on 48 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=fgh,c^2=fi,d^2=hi,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=ef,d^c=dfg

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4908931);