#4448034

order	512
nilpotency class	2
isoclinism	isoclinic class of #4417862
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

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

1435 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fg,c^2=f,d^2=fh,e^2=fhi,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch,e^d=ef

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, 4355230);