#4933507

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄²

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

2290 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)(6,7)(8,12)(14,16)(15,19)(23,26)(24,28);(5,6,9,7)(8,11,12,10)(13,14,18,16)(15,20,19,17)(21,22,25,27)(23,24,26,28);(5,6,9,7)(8,10,12,11)(13,15)(14,17)(16,20)(18,19)(21,23)(22,24)(25,26)(27,28);(1,2)(3,4)(5,7,9,6)(8,11,12,10)(13,16,18,14)(15,20,19,17)(21,24)(22,26)(23,27)(25,28);(5,8,9,12)(6,10,7,11)(13,17,18,20)(14,19,16,15)

Polycyclic presentation:

2^9|b^2=g,c^2=fh,d^2=f,e^2=f,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, 4840703);