#42076

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	#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

801 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=f,c^2=fg,d^2=fh,e^2=g,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×#42076, #7640413, #7640512, …
maximal normal subgroup of	C₂×#42076, #7640413, #7640512, …
maximal quotient of	#544818, #545269, #545762, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35064);