#57938

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

647 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)(3,6)(4,9)(5,10)(7,11)(8,13)(12,14)(15,16)(17,18,22,20)(19,23,25,27)(21,28,29,24)(26,32,31,30)(33,34)(35,37)(36,40)(38,46)(39,42)(41,47)(43,48)(44,45);(2,3)(4,11)(8,10)(12,16)(18,20)(19,25)(21,29)(30,32)(33,35,39,36)(34,40,42,37)(38,43,47,44)(41,48,46,45);(1,3,6,2)(4,9,11,7)(5,10,13,8)(12,15,16,14)(17,19)(18,23)(20,27)(21,26)(22,25)(24,30)(28,32)(29,31)(33,36,39,35)(34,40,42,37)(38,44,47,43)(41,48,46,45);(1,4)(2,7)(3,9)(5,12)(6,11)(8,14)(10,15)(13,16)(17,20,22,18)(19,23,25,27)(21,28,29,24)(26,30,31,32)(33,37,39,40)(34,35,42,36)(38,45,47,48)(41,43,46,44);(1,5)(2,8)(3,10)(4,12)(6,13)(7,14)(9,15)(11,16)(17,21,22,29)(18,24,20,28)(19,26,25,31)(23,30,27,32)(33,38,39,47)(34,41,42,46)(35,43,36,44)(37,45,40,48)

Polycyclic presentation:

2^8|a^2=gh,b^2=fh,c^2=fgh,d^2=fg,e^2=fg,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×#57938, #8101368, #8101956, …
maximal normal subgroup of	C₂×#57938, #8101368, #8101956, …
maximal quotient of	#586336, #586470, #587262, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50926);