#2836535

order	512
nilpotency class	2
isoclinism	isoclinic class of #2805613
automorphism	Aut(G) order 1048576
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	C₂²×#120	C₂²
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

2786 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)(11,13)(12,14)(15,16)(17,18,20,22)(19,23,24,21)(26,29)(27,28);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,18,20,22)(19,21,24,23)(25,26)(27,31)(28,32)(29,30);(17,18)(19,21)(20,22)(23,24)(25,27,30,28)(26,31,29,32);(1,3,4,8)(2,6,5,7)(9,11,12,16)(10,13,14,15)(17,18)(19,21)(20,22)(23,24)(25,28,30,27)(26,31,29,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19)(18,21)(20,24)(22,23)(25,29,30,26)(27,32,28,31)

Polycyclic presentation:

2^9|a^2=h,b^2=fi,c^2=gh,d^2=f,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^b=ef,e^d=ef

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72328607, #72328651, #72328704, …
maximal normal subgroup of	#72328607, #72328651, #72328704, …
maximal quotient of	#72330768

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2743731);