#35499

order	256
nilpotency class	2
isoclinism	isoclinic class of #35346
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)(3,4,7,6)(9,10,13,12)(11,15,16,14)(17,18,22,20)(19,25,26,23)(21,24,29,28)(27,31,32,30);(1,3,5,7)(2,6,8,4)(9,11,13,16)(10,14,12,15)(19,26)(21,29)(23,25)(24,28);(1,2)(3,6)(4,7)(5,8)(9,10,13,12)(11,14,16,15)(17,19,22,26)(18,23,20,25)(21,27,29,32)(24,30,28,31);(9,10,13,12)(11,14,16,15)(17,20,22,18)(19,23,26,25)(21,28,29,24)(27,30,32,31);(1,4,5,6)(2,7,8,3)(9,12,13,10)(11,15,16,14)(17,21)(18,24)(19,27)(20,28)(22,29)(23,30)(25,31)(26,32)

Polycyclic presentation:

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

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#35499, #7628872, #7628875, …
maximal normal subgroup of	C₂×#35499, #7628872, #7628875, …
maximal quotient of	#515464, #515468, #515472, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 28487);