#62972

order	256
nilpotency class	2
isoclinism	isoclinic class of #62972
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

3411 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 32

A minimal faithful permutation representation — 6 generators on 32 points:

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

Polycyclic presentation:

2^8|b^a=bg,c^a=ch,d^a=dh,f^a=fg,d^b=dh,e^b=eg,f^b=fh,d^c=dg,e^c=eh

Conjugacy classes & character table

67 conjugacy classes — view the full character table.

Relationships

derived subgroup of	#62972⋊C₇
Fitting subgroup of	#62972⋊C₃, #62972⋊C₇
maximal subgroup of	#347034, #347418, C₂×#62972, …
maximal normal subgroup of	#347034, #347418, C₂×#62972, …
maximal quotient of	#7627617, #7627641, #7627673, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 55960);