#4308371

order	512
nilpotency class	2
isoclinism	isoclinic class of #4307937
automorphism	Aut(G) order 8388608
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	C₂⁵	C₂⁴
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2818 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)(17,18,21,22)(19,23,26,29)(20,27,28,24)(25,31,32,30);(1,2,4,3)(9,10)(11,14)(12,13)(15,16)(18,22)(23,29)(24,27)(30,31);(5,6)(7,8)(9,10)(11,14)(12,13)(15,16)(17,18)(19,23)(20,24)(21,22)(25,30)(26,29)(27,28)(31,32);(1,3,4,2)(9,11,12,15)(10,14,13,16)(17,19,21,26)(18,23,22,29)(20,25,28,32)(24,30,27,31);(17,20,21,28)(18,24,22,27)(19,25,26,32)(23,30,29,31)

Polycyclic presentation:

2^9|a^2=h,b^2=i,d^2=f,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, 4215567);