#35924

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂³	C₂⁵
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1150 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 5 generators on 24 points:

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

Polycyclic presentation:

2^8|b^2=f,c^2=g,d^2=g,e^2=f,b^a=bf,c^a=cg,c^b=ch,e^d=eh

Conjugacy classes & character table

64 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#35924⋊C₃, #72260639, #72260656, …
maximal subgroup of	#132305, #253795, #253797, …
maximal normal subgroup of	#132305, #253795, #253797, …
maximal quotient of	#517981, #518004, #518016, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 28912);