#40938

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	#392	C₂²
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

844 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)(11,14)(12,13)(15,16)(17,18,22,25)(19,23,27,21)(20,28,30,24)(26,29,32,31);(1,2)(3,7)(4,5)(6,8)(9,10)(11,15)(12,13)(14,16)(17,19,22,27)(18,21,25,23)(20,26,30,32)(24,29,28,31);(2,5)(6,7)(10,13)(14,15)(17,19)(18,23)(20,26)(21,25)(22,27)(24,31)(28,29)(30,32);(1,2)(3,6)(4,5)(7,8)(9,10)(11,14)(12,13)(15,16)(17,20,22,30)(18,24,25,28)(19,26,27,32)(21,29,23,31);(1,3)(2,6)(4,8)(5,7)(9,11,12,16)(10,14,13,15)(17,21,22,23)(18,19,25,27)(20,29,30,31)(24,26,28,32)

Polycyclic presentation:

2^8|a^2=f,b^2=f,d^2=f,e^2=fg,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×#40938, #7639510, #7639585, …
maximal normal subgroup of	C₂×#40938, #7639510, #7639585, …
maximal quotient of	#544795, #545249, #545861, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 33926);