#36054

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
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	C₂×#121	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

832 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=fgh,b^2=fg,c^2=f,d^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#36054, #7629478, #7629511, …
maximal normal subgroup of	C₂×#36054, #7629478, #7629511, …
maximal quotient of	#517954, #517956, #517957, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29042);