#57884

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

694 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=h,b^2=gh,c^2=h,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×#57884, #8101754, #8101959, …
maximal normal subgroup of	C₂×#57884, #8101754, #8101959, …
maximal quotient of	#586658, #586862, #587326, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 50872);