#41323

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

736 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:

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

Polycyclic presentation:

2^8|a^2=f,b^2=g,c^2=g,d^2=fg,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×#41323, #7638930, #7639599, …
maximal normal subgroup of	C₂×#41323, #7638930, #7639599, …
maximal quotient of	#544963, #545333, #545971, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34311);