#2594065

order	512
nilpotency class	2
isoclinism	isoclinic class of #2076493
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁵
prime radical	#14715	C₂
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

3666 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)(3,5)(4,6)(7,8)(9,10)(11,16)(12,14)(13,15)(17,18)(19,21)(20,24)(22,23)(26,30)(27,28);(2,3)(6,7)(9,11)(10,12)(13,14)(15,16)(18,23)(19,20)(25,26)(27,32)(28,31)(29,30);(4,8)(6,7)(11,14)(12,16)(17,19,22,20)(18,21,23,24)(25,27,29,28)(26,31,30,32);(1,3,5,2)(4,7,8,6)(9,10)(11,16)(12,14)(13,15)(17,20,22,19)(18,21,23,24)(25,28,29,27)(26,31,30,32);(1,4,5,8)(2,6,3,7)(9,12,13,16)(10,14,15,11)(17,21)(18,19)(20,23)(22,24)(25,26)(27,32)(28,31)(29,30)

Polycyclic presentation:

2^9|c^2=fhi,d^2=i,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	#72262805, #72262817, #72262819, …
maximal normal subgroup of	#72262805, #72262817, #72262819, …
maximal quotient of	#72328717, #72328869, #72339677

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2501261);