#51413

order	256
nilpotency class	2
isoclinism	isoclinic class of #47826
automorphism	Aut(G) = C₂¹⁵ (order 32768) · Inn(G) = C₂⁵ · Out(G) = C₂¹⁰
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁸

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

950 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 36 points:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=f,c^2=h,d^2=h,e^2=f,b^a=bf,c^a=cg,e^a=ef,c^b=ch,d^b=dfh,e^b=eh,d^c=dh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#51413, #7707039, #7707041, …
maximal normal subgroup of	C₂×#51413, #7707039, #7707041, …
maximal quotient of	#560360, #560390, #561455, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 44401);