#4934756

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

2677 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

(2,4)(6,10)(7,11)(13,14)(15,19)(16,20)(17,18)(23,29)(24,31)(27,32)(28,33);(5,6)(7,8)(9,10)(11,12)(13,14,17,18)(15,19,20,16)(22,26)(27,32)(28,33)(34,36);(6,10)(7,11)(13,15)(14,16)(17,20)(18,19)(21,22)(23,27)(24,28)(25,26)(29,32)(30,34)(31,33)(35,36);(1,2)(3,4)(5,7,9,11)(6,8,10,12)(13,16)(14,15)(17,19)(18,20)(21,23,25,29)(22,27,26,32)(24,30,31,35)(28,34,33,36);(5,8)(6,11)(7,10)(9,12)(13,14)(15,16)(17,18)(19,20)(21,24)(22,28)(23,30)(25,31)(26,33)(27,34)(29,35)(32,36)

Polycyclic presentation:

2^9|b^2=g,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, 4841952);