#41047

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	C₂²×#41	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

842 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)(11,13)(12,14)(15,16)(17,18,22,25)(19,20,27,23)(21,24,29,31)(26,28,32,30);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,22,27)(18,23,25,20)(21,26,29,32)(24,30,31,28);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,19)(18,20)(21,26)(22,27)(23,25)(24,28)(29,32)(30,31);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,20)(18,19)(21,28)(22,23)(24,26)(25,27)(29,30)(31,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,21,22,29)(18,24,25,31)(19,26,27,32)(20,28,23,30)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=g,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×#41047, #7639278, #7639587, …
maximal normal subgroup of	C₂×#41047, #7639278, #7639587, …
maximal quotient of	#544815, #545265, #545759, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34035);