#665056

order	512
nilpotency class	2
isoclinism	isoclinic class of #647656
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#121	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2392 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)(6,7)(9,10)(11,13)(12,15)(14,16)(17,18,23,22)(19,20,21,24)(25,26)(27,29)(28,30)(31,32);(1,2,5,3)(4,7,8,6)(9,11,14,12)(10,15,16,13)(17,19,23,21)(18,24,22,20)(26,27)(28,32);(9,11,14,12)(10,13,16,15)(17,20)(18,21)(19,22)(23,24)(25,27,29,26)(28,31,32,30);(1,3,5,2)(4,7,8,6)(9,12,14,11)(10,15,16,13)(17,21,23,19)(18,24,22,20);(1,4,5,8)(2,6,3,7)(9,13,14,15)(10,11,16,12)(17,22,23,18)(19,24,21,20)(25,28,29,32)(26,30,27,31)

Polycyclic presentation:

2^9|a^2=gh,b^2=fg,c^2=fghi,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=di,c^b=ch,e^b=ef

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, 572252);