#46611

order	256
nilpotency class	2
isoclinism	isoclinic class of #46350
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

817 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)(11,14)(12,13)(15,16)(17,18,23,21)(19,26,20,24)(22,29,30,25)(27,31,28,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,23,20)(18,24,21,26)(22,27,30,28)(25,31,29,32);(2,5)(6,7)(10,13)(14,15)(17,20,23,19)(18,24,21,26)(22,28,30,27)(25,31,29,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,21,23,18)(19,24,20,26)(22,29,30,25)(27,31,28,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,22)(18,25)(19,27)(20,28)(21,29)(23,30)(24,31)(26,32)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=f,d^2=f,e^2=g,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=df,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#46611, #7700435, #7700572, …
maximal normal subgroup of	C₂×#46611, #7700435, #7700572, …
maximal quotient of	#576969, #577020, #577104, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 39599);