#4308377

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 16777216
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

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

2762 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:

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=g,d^2=g,e^2=h,b^a=bf,c^a=cg,d^a=di,e^a=eh,c^b=ch

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 4215573);