#4307440

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄×#392
automorphism	Aut(G) order 2097152
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

1940 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,5,6)(3,4,7,8)(9,10)(11,12)(13,15)(14,16)(17,18)(19,21)(20,23)(22,24)(25,26)(27,31)(28,30)(29,32);(2,6)(4,8)(9,11)(10,15)(12,16)(13,14)(17,19,22,20)(18,23,24,21)(25,26)(27,29)(28,30)(31,32);(2,6)(4,8)(10,16)(11,13)(17,19,22,20)(18,21,24,23)(25,26,28,30)(27,29,32,31);(1,3,5,7)(2,4,6,8)(9,12,14,15)(10,11,16,13)(17,20,22,19)(18,23,24,21)(25,27)(26,29)(28,32)(30,31);(1,4)(2,7)(3,6)(5,8)(9,13,14,11)(10,15,16,12)(17,21,22,23)(18,19,24,20)(25,26)(27,29)(28,30)(31,32)

Polycyclic presentation:

2^9|a^2=gh,b^2=fgi,c^2=fg,d^2=f,e^2=fgh,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,e^c=eg

Conjugacy classes & character table

110 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4214636);