#52832

order	256
nilpotency class	2
isoclinism	isoclinic class of #52308
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	C₂⁴	C₂⁴
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

629 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 48 points:

(1,2,5,6)(3,7,9,10)(4,12,13,8)(11,16,15,14)(17,18)(19,23)(20,22)(21,24)(25,26)(27,30)(28,29)(31,32)(36,44)(39,46)(42,48)(45,47);(2,6)(3,9)(8,12)(11,15)(18,20)(19,25)(24,28)(27,31)(33,34,37,40)(35,38,43,41)(36,39,44,46)(42,45,48,47);(1,2)(3,10)(4,8)(5,6)(7,9)(11,16)(12,13)(14,15)(17,19,22,25)(18,23,20,26)(21,27,29,31)(24,30,28,32)(33,34)(35,41)(36,39)(37,40)(38,43)(42,47)(44,46)(45,48);(1,3)(2,7)(4,11)(5,9)(6,10)(8,14)(12,16)(13,15)(17,20,22,18)(19,26,25,23)(21,28,29,24)(27,32,31,30)(33,35)(34,38)(36,42)(37,43)(39,45)(40,41)(44,48)(46,47);(1,4,5,13)(2,8,6,12)(3,11,9,15)(7,14,10,16)(17,21,22,29)(18,24,20,28)(19,27,25,31)(23,30,26,32)(33,36)(34,39)(35,42)(37,44)(38,45)(40,46)(41,47)(43,48)

Polycyclic presentation:

2^8|a^2=g,b^2=fh,c^2=fg,d^2=fg,e^2=f,b^a=bf,c^a=cg,d^a=dfg,e^a=efgh,c^b=ch,d^b=df,d^c=dfgh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#52832, #8092603, #8092703, …
maximal normal subgroup of	C₂×#52832, #8092603, #8092703, …
maximal quotient of	#582314, #582519, #582565, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 45820);