#4216015

order	512
nilpotency class	2
isoclinism	isoclinic class of #4131229
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15320	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2233 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:

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

Polycyclic presentation:

2^9|a^2=hi,b^2=fg,c^2=ghi,e^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,e^b=eg,d^c=dg,e^c=ef,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4123211);