#63016

order	256
nilpotency class	2
isoclinism	isoclinic class of #63011
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

3534 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 6 generators on 24 points:

(1,2)(3,6)(4,7)(5,8)(9,10,15,19)(11,17,12,16)(13,14,18,24)(20,23,22,21);(1,2)(3,6)(4,7)(5,8)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23);(1,3)(2,5)(4,8)(6,7)(9,11,15,12)(10,16,19,17)(13,20,18,22)(14,21,24,23);(9,12,15,11)(10,17,19,16)(13,22,18,20)(14,23,24,21);(1,3)(2,6)(4,8)(5,7)(9,13)(10,14)(11,20)(12,22)(15,18)(16,21)(17,23)(19,24);(1,2)(3,5)(4,7)(6,8)(9,14)(10,18)(11,21)(12,23)(13,19)(15,24)(16,22)(17,20)

Polycyclic presentation:

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

Conjugacy classes & character table

76 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#63016⋊C₃, #72268256, #72271703, …
maximal subgroup of	#346937, #347191, #347218, …
maximal normal subgroup of	#346937, #347191, #347218, …
maximal quotient of	#7627216, #7627224, #7627225, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56004);