#41302

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

783 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)(11,16,13,15)(18,22)(19,25)(24,29)(26,31);(1,2)(3,7)(4,5)(6,8)(11,13)(15,16)(17,18,21,22)(19,23,25,28)(20,24,27,29)(26,30,31,32);(2,5)(6,7)(9,11,14,13)(10,15,12,16)(17,18)(19,23)(20,24)(21,22)(25,28)(26,30)(27,29)(31,32);(1,2)(3,6)(4,5)(7,8)(9,12,14,10)(11,15,13,16)(17,19)(18,23)(20,26)(21,25)(22,28)(24,30)(27,31)(29,32);(1,3)(2,6)(4,8)(5,7)(9,13,14,11)(10,15,12,16)(17,20,21,27)(18,24,22,29)(19,26,25,31)(23,30,28,32)

Polycyclic presentation:

2^8|a^2=fg,b^2=f,c^2=fg,d^2=fg,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×#41302, #7639104, #7639596, …
maximal normal subgroup of	C₂×#41302, #7639104, #7639596, …
maximal quotient of	#544945, #545329, #545961, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 34290);