#63103

order	256
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁷ · Schur multiplier (H₂) = C₂²⁰

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄	C₂⁶
commutator, Frattini, socle, cosocle	C₂	C₂⁷
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₂ ▷ C₁

29726 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 7 generators on 32 points:

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

Polycyclic presentation:

2^8|g^2=h,b^a=bh,f^a=fh,e^b=eh,d^c=dh

Conjugacy classes & character table

130 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11710936, #11710937, #11710938, …
maximal subgroup of	#392140, #6370739, #6370751, …
maximal normal subgroup of	#392140, #6370739, #6370751, …
maximal quotient of	#10585883, C₄×H(7,𝔽₂), C₄×2₋¹⁺⁶, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56091);