#12031

order	256
nilpotency class	4
isoclinism	isoclinic class of #12028
automorphism	Aut(G) order 4096
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂×#106
commutator	C₂×C₈	C₂²×C₄
Frattini, cosocle	C₄×C₈	C₂³
prime radical	C₂³	#106
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂×C₄² ◁ C₂×C₄×C₈ ◁ G
LECS	G ▷ C₄×C₈ ▷ C₂×C₄ ▷ C₂ ▷ C₁

100 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 48 points:

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

Polycyclic presentation:

2^8|a^2=e,b^2=d,c^2=h,d^2=f,e^2=g,f^2=h,b^a=bd,d^a=df,f^a=fh,c^b=cgh,e^b=eh

Conjugacy classes & character table

46 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#145213, #145214, #145215, …
maximal normal subgroup of	#145213, #145214, #145215, …
maximal quotient of	#126711, #129127, #134690, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 5019);