#4973765

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, cosocle	C₂⁴	C₂⁵
prime radical	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2178 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:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=gh,c^2=i,d^2=gh,e^2=gh,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, 4880961);