#1586434

order	512
nilpotency class	2
isoclinism	isoclinic class of #1506685
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#15257	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2856 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72327622, #72327672, #72327725
maximal normal subgroup of	#72327622, #72327672, #72327725
maximal quotient of	#72328945

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 1493630);