#2358058

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁴×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3216 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:

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

Polycyclic presentation:

2^9|a^2=fgh,b^2=hi,c^2=fh,d^2=fghi,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	#72263179, #72263207, #72263222, …
maximal normal subgroup of	#72263179, #72263207, #72263222, …
maximal quotient of	#72330936

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2265254);