#4949441

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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2652 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)(5,6,8,9)(7,11,12,10)(13,14,17,15)(16,19,20,18)(23,27)(26,28);(6,9)(10,11)(13,14,17,15)(16,18,20,19)(22,25)(26,28);(2,3)(7,12)(10,11)(14,15)(18,19)(21,22)(23,26)(24,25)(27,28);(1,3,4,2)(5,7)(6,10)(8,12)(9,11)(13,15,17,14)(16,19,20,18)(21,23)(22,26)(24,27)(25,28);(5,6)(7,10)(8,9)(11,12)(13,16)(14,18)(15,19)(17,20)(21,23)(22,26)(24,27)(25,28)

Polycyclic presentation:

2^9|a^2=fg,b^2=fhi,d^2=g,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, 4856637);