#2172325

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
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	C₂³×#41	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2594 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:

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

Polycyclic presentation:

2^9|a^2=ghi,b^2=hi,d^2=f,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dfg,e^c=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2079521);