#36043

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	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

814 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:

(2,3)(6,7)(9,10,13,16)(11,14,12,15)(17,18)(19,24)(20,25)(21,26)(22,23)(27,29)(28,32)(30,31);(1,2,5,3)(4,6,8,7)(9,10)(11,14)(12,15)(13,16)(18,23)(19,25)(26,31)(27,32);(1,3,5,2)(4,7,8,6)(9,11,13,12)(10,14,16,15)(17,19,22,25)(18,24,23,20)(21,27,30,32)(26,29,31,28);(1,4,5,8)(2,6,3,7)(9,12,13,11)(10,15,16,14)(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=g,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×#36043, #7629478, #7629501, …
maximal normal subgroup of	C₂×#36043, #7629478, #7629501, …
maximal quotient of	#517893, #517895, #517896, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29031);