#40726

order	256
nilpotency class	2
isoclinism	isoclinic class of #40484
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

799 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:

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

Polycyclic presentation:

2^8|a^2=g,b^2=f,c^2=f,d^2=g,b^a=bf,c^a=cg,d^a=df,c^b=ch,e^b=efh,d^c=dfh

Conjugacy classes & character table

52 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#40726, #7638843, #7639578, …
maximal normal subgroup of	C₂×#40726, #7638843, #7639578, …
maximal quotient of	#544617, #545147, #545611, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 33714);