#2471118

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
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

2246 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:

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

Polycyclic presentation:

2^9|a^2=fh,c^2=g,e^2=hi,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.

Relationships

maximal subgroup of	#72304048, #72304078, #72304113
maximal normal subgroup of	#72304048, #72304078, #72304113

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2378314);