#25809

order	256
nilpotency class	3
isoclinism	isoclinic class of #24292
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	D₄²
commutator, Frattini, cosocle	C₂²×C₄	C₂⁴
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄ ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

707 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^8|b^2=e,c^2=f,d^2=eh,e^2=h,g^2=h,b^a=be,c^a=cf,d^a=de,e^a=eh,c^b=cg,g^b=gh,d^c=dh,g^c=gh

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#25809, #6443285, #6443288, …
maximal normal subgroup of	C₂×#25809, #6443285, #6443288, …
maximal quotient of	#299302, #299306, #299308, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 18797);