#41829

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×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

732 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=fg,d^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×#41829, #7640012, #7640247, …
maximal normal subgroup of	C₂×#41829, #7640012, #7640247, …
maximal quotient of	#544947, #545326, #545963, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34817);