#42106

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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

798 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)(3,6)(4,5)(7,8)(17,18,22,25)(19,26,20,23)(21,24,29,32)(27,31,28,30);(1,2)(3,7)(4,5)(6,8)(9,10)(11,12)(13,14)(15,16)(17,19,22,20)(18,23,25,26)(21,28,29,27)(24,31,32,30);(2,5)(6,7)(10,14)(11,15)(17,20,22,19)(18,23,25,26)(21,28,29,27)(24,30,32,31);(1,2)(3,6)(4,5)(7,8)(9,11,13,15)(10,12,14,16)(17,18)(19,23)(20,26)(21,24)(22,25)(27,30)(28,31)(29,32);(1,3,4,8)(2,6,5,7)(9,12)(10,15)(11,14)(13,16)(17,21,22,29)(18,24,25,32)(19,27,20,28)(23,30,26,31)

Polycyclic presentation:

2^8|a^2=f,b^2=f,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×#42106, #7640431, #7640473, …
maximal normal subgroup of	C₂×#42106, #7640431, #7640473, …
maximal quotient of	#544988, #545355, #545892, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35094);