#2227608

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	C₂×#392	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2260 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)(4,8)(5,6)(9,10,14,15)(11,16,13,12)(17,18,23,24)(19,21,22,20)(26,30)(28,32);(1,3)(2,4)(5,8)(6,7)(9,11,14,13)(10,12,15,16)(17,19,23,22)(18,20,24,21)(26,30)(27,31);(2,6)(4,7)(10,15)(11,13)(17,20,23,21)(18,19,24,22)(25,26)(27,28)(29,30)(31,32);(1,3)(2,4)(5,8)(6,7)(9,12,14,16)(10,11,15,13)(17,21,23,20)(18,19,24,22)(25,27)(26,28)(29,31)(30,32);(1,4)(2,3)(5,7)(6,8)(9,13,14,11)(10,16,15,12)(17,22,23,19)(18,20,24,21)(25,28)(26,27)(29,32)(30,31)

Polycyclic presentation:

2^9|a^2=fgh,b^2=fgh,c^2=ghi,d^2=fgh,e^2=fgh,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, 2134804);