#712927

order	512
nilpotency class	2
isoclinism	isoclinic class of #686200
automorphism	?
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2644 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=fhi,c^2=fi,d^2=fg,e^2=fg,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef,d^c=df

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 620123);