#33474

order	256
nilpotency class	2
isoclinism	isoclinic class of #120
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂³×C₈ · Schur multiplier (H₂) = C₂⁴

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₁₆	C₂³
commutator, socle	C₂²	C₂³×C₈
Frattini, cosocle	C₂×C₈	C₂⁴
prime radical	C₂³	C₂²×C₈
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂×C₁₆ ◁ G
LECS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁

253 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 40
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 40 points:

(1,2)(3,8)(4,6)(5,7)(9,10,12,17,13,18,22,30,14,16,23,28,24,29,34,38)(11,15,19,26,20,27,31,37,21,25,32,35,33,36,39,40);(1,3,5,4)(2,6,7,8);(1,4,5,3)(2,6,7,8)(9,11,12,19,13,20,22,31,14,21,23,32,24,33,34,39)(10,15,17,26,18,27,30,37,16,25,28,35,29,36,38,40);(10,16)(15,25)(17,28)(18,29)(26,35)(27,36)(30,38)(37,40)

Polycyclic presentation:

2^8|a^2=f,b^2=e,c^2=ef,f^2=g,g^2=h,b^a=be,d^a=dh

Conjugacy classes & character table

112 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#128510, #128511, #128512, …
maximal normal subgroup of	#128510, #128511, #128512, …
maximal quotient of	#493116, #493118, #493258, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 26462);