#39099

order	256
nilpotency class	2
isoclinism	isoclinic class of #37644
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

737 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=f,d^2=fg,e^2=gh,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#39099, #7636480, #7636536, …
maximal normal subgroup of	C₂×#39099, #7636480, #7636536, …
maximal quotient of	#521581, #522279, #522834, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 32087);