#63048

order	256
nilpotency class	2
isoclinism	isoclinic class of #63041
automorphism	?
homology	Abelianization (H₁) = C₂⁶ · Schur multiplier (H₂) = C₂¹³

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂²	C₂⁶
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ G
LECS	G ▷ C₂² ▷ C₁

3456 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 40 points:

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

Polycyclic presentation:

2^8|a^2=g,b^2=g,e^2=h,f^2=h,b^a=bg,c^a=ch,d^a=dg,c^b=cgh,d^b=dh,f^e=fh

Conjugacy classes & character table

70 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#11710852, #72260605, C₂×#11710852, …
Fitting subgroup of	#11710852, #11710853, #11710854, …
maximal subgroup of	#347435, #347455, #347475, …
maximal normal subgroup of	#347435, #347455, #347475, …
maximal quotient of	#7627406, #7627427, #7627433, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56036);