#63076

order	256
nilpotency class	2
isoclinism	isoclinic class of H(7,𝔽₂)
automorphism	?
homology	Abelianization (H₁) = C₂⁵×C₄ · Schur multiplier (H₂) = C₂¹⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₄	C₂⁶
commutator, socle	C₂	C₂⁵×C₄
prime radical	C₂×#585	C₂
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₄ ◁ G
LECS	G ▷ C₄ ▷ C₂ ▷ C₁

5398 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:

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

Polycyclic presentation:

2^8|a^2=g,g^2=h,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	#11710872, #11710873, #63076⋊C₅, …
maximal subgroup of	#392597, #392612, #392615, …
maximal normal subgroup of	#392597, #392612, #392615, …
maximal quotient of	#10574035, #10574037, #10574081, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 56064);