#36072

order	256
nilpotency class	2
isoclinism	isoclinic class of #35680
automorphism	Aut(G) order 65536
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
prime radical	C₂²×D₄	C₂³
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ G
LECS	G ▷ C₂³ ▷ C₁

796 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

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

Polycyclic presentation:

2^8|a^2=fh,b^2=fg,c^2=fg,d^2=fg,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

maximal subgroup of	#132441, #253640, #253642, …
maximal normal subgroup of	#132441, #253640, #253642, …
maximal quotient of	#517830, #517832, #517841, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 29060);