#42199

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

680 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=fg,c^2=g,d^2=fh,e^2=f,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#42199, #7640694, #7640701, …
maximal normal subgroup of	C₂×#42199, #7640694, #7640701, …
maximal quotient of	#544536, #545118, #545398, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 35187);