#33999

order	256
nilpotency class	2
isoclinism	isoclinic class of H(5,𝔽₂)
automorphism	Aut(G) order 4718592
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂⁴
commutator	C₂	C₂³×C₄²
Frattini, socle, cosocle	C₂³	C₂⁵
prime radical	C₂²×#41	C₂²
LCS	G ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂³ ▷ C₁

1799 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

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

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

Polycyclic presentation:

2^8|a^2=g,b^2=h,c^2=g,e^2=f,b^a=bf,d^a=df,c^b=cf

Conjugacy classes & character table

136 conjugacy classes — view the full character table.

Relationships

Fitting subgroup of	#11709754, #11709755, #11709756, …
maximal subgroup of	#105680, #107770, #180860, …
maximal normal subgroup of	#105680, #107770, #180860, …
maximal quotient of	#513329, C₄×#2275, C₄×#2276, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26987);