#2626870

order	512
nilpotency class	2
isoclinism	isoclinic class of #2601293
automorphism	Aut(G) order 1048576
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

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

1995 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:

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

Polycyclic presentation:

2^9|a^2=fgi,b^2=ghi,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=di,e^a=eg,c^b=ch,d^b=dg,e^d=ef

Conjugacy classes & character table

74 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72266510, #72266512, #72266520, …
maximal normal subgroup of	#72266510, #72266512, #72266520, …
maximal quotient of	#72334867

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 2534066);