#41039

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	#2813	C₂
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

855 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)(11,13)(12,14)(15,16)(17,18,23,28)(19,25,20,24)(21,22,27,26)(29,30,31,32);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,14)(13,16)(17,19,23,20)(18,24,28,25)(21,29,27,31)(22,32,26,30);(2,5)(6,7)(9,10,12,14)(11,13,16,15)(17,20,23,19)(18,25,28,24)(21,29,27,31)(22,32,26,30);(1,2)(3,6)(4,5)(7,8)(9,10)(11,13)(12,14)(15,16)(17,21,23,27)(18,26,28,22)(19,29,20,31)(24,30,25,32);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,13,14,15)(17,22,23,26)(18,27,28,21)(19,30,20,32)(24,29,25,31)

Polycyclic presentation:

2^8|a^2=f,b^2=f,c^2=g,d^2=f,e^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×#41039, #7639048, #7639587, …
maximal normal subgroup of	C₂×#41039, #7639048, #7639587, …
maximal quotient of	#544843, #545274, #545819, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34027);