#55392

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

689 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:

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

Polycyclic presentation:

2^8|a^2=fg,b^2=gh,c^2=fg,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×#55392, #8102583, #8102631, …
maximal normal subgroup of	C₂×#55392, #8102583, #8102631, …
maximal quotient of	#586274, #586850, #587077, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 48380);