#4954769

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
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	C₂⁶	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2472 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,c^2=fh,d^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, 4861965);