#42290

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

716 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)(3,4)(5,6)(7,8)(9,10,13,16)(11,14,12,15)(17,18,22,19)(20,25,28,23)(21,24,29,26)(27,31,32,30);(2,6)(4,7)(9,11,13,12)(10,14,16,15)(18,19)(23,25)(24,26)(30,31);(1,3,5,8)(2,4,6,7)(9,10)(11,14)(12,15)(13,16)(17,19,22,18)(20,25,28,23)(21,26,29,24)(27,31,32,30);(1,3)(2,7)(4,6)(5,8)(9,11,13,12)(10,14,16,15)(17,20,22,28)(18,23,19,25)(21,27,29,32)(24,30,26,31);(1,4,5,7)(2,3,6,8)(9,12,13,11)(10,15,16,14)(17,21,22,29)(18,24,19,26)(20,27,28,32)(23,30,25,31)

Polycyclic presentation:

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

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#42290, #7640638, #7640656, …
maximal normal subgroup of	C₂×#42290, #7640638, #7640656, …
maximal quotient of	#544987, #545357, #545891, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35278);