#58509

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

643 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:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=gh,c^2=fg,d^2=h,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×#58509, #8104319, #8104434, …
maximal normal subgroup of	C₂×#58509, #8104319, #8104434, …
maximal quotient of	#586273, #586636, #586749, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 51497);