#1120901

order	512
nilpotency class	2
isoclinism	isoclinic class of #1082705
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

2068 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1028097);