#4933007

order	512
nilpotency class	2
isoclinism	isoclinic class of #4932388
automorphism	Aut(G) order 2097152
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15412	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3067 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^9|b^2=h,c^2=fg,d^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, 4840203);