#54692

order	256
nilpotency class	2
isoclinism	isoclinic class of #53390
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

645 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2)(3,7)(4,12)(5,6)(8,13)(9,10)(11,16)(14,15)(17,18,22,23)(19,20,24,27)(21,29,30,25)(26,32,31,28)(34,35)(36,43)(37,45)(46,47);(2,6)(3,9)(4,13)(14,16)(18,23)(20,27)(25,29)(28,32)(33,34,38,35)(36,41,43,39)(37,42,45,40)(44,46,48,47);(1,3,5,9)(2,7,6,10)(4,11,13,15)(8,14,12,16)(17,19,22,24)(18,20,23,27)(21,26,30,31)(25,28,29,32)(33,35,38,34)(36,39,43,41)(37,42,45,40)(44,46,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,20)(18,24)(19,23)(21,28)(22,27)(25,31)(26,29)(30,32)(33,36,38,43)(34,39,35,41)(37,44,45,48)(40,46,42,47);(1,4)(2,8)(3,11)(5,13)(6,12)(7,14)(9,15)(10,16)(17,21)(18,25)(19,26)(20,28)(22,30)(23,29)(24,31)(27,32)(33,37,38,45)(34,40,35,42)(36,44,43,48)(39,46,41,47)

Polycyclic presentation:

2^8|a^2=fgh,b^2=g,c^2=f,d^2=g,e^2=g,b^a=bf,c^a=cg,d^a=dg,e^a=ef,c^b=ch,d^b=df,e^b=egh,d^c=dgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#54692, #8103696, #8103787, …
maximal normal subgroup of	C₂×#54692, #8103696, #8103787, …
maximal quotient of	#586394, #586626, #586942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 47680);