#865712

order	512
nilpotency class	2
isoclinism	isoclinic class of #845297
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶

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

1701 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)(4,8)(10,13)(14,16)(17,18,22,20)(19,21,24,23)(25,26)(27,32)(28,30)(29,31);(1,2,5,3)(4,6,8,7)(10,13)(14,16)(17,19,22,24)(18,23,20,21)(26,28)(31,32);(9,10)(11,14)(12,13)(15,16)(17,18,22,20)(19,23,24,21)(25,27,30,29)(26,31,28,32);(1,3,5,2)(4,7,8,6)(9,10)(11,14)(12,13)(15,16)(17,20,22,18)(19,21,24,23)(25,28,30,26)(27,32,29,31);(1,4)(2,6)(3,7)(5,8)(9,11,12,15)(10,14,13,16)(17,21)(18,19)(20,24)(22,23)(25,29,30,27)(26,32,28,31)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fgh,c^2=fgi,d^2=f,e^2=g,b^a=bf,c^a=cg,d^a=di,e^a=ef,c^b=ch,d^b=dg

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, 772908);