#5480386

order	512
nilpotency class	2
isoclinism	isoclinic class of #5473759
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1830 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 28 points:

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fg,c^2=h,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=efh,c^b=ch,e^b=efg,d^c=dg,e^c=ef

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 5387582);