#38491

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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=fg,c^2=g,d^2=h,e^2=f,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×#38491, #7636132, #7636312, …
maximal normal subgroup of	C₂×#38491, #7636132, #7636312, …
maximal quotient of	#521815, #522389, #523036, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 31479);