#36071

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

782 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:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=fg,c^2=fg,d^2=g,e^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×#36071, #7629540, #7629545, …
maximal normal subgroup of	C₂×#36071, #7629540, #7629545, …
maximal quotient of	#517831, #517832, #517833, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29059);