#30487

order	256
nilpotency class	3
isoclinism	isoclinic class of #30484
automorphism	Aut(G) order 8192
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁴×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

508 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

40 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6632111, #6668398, #6668399, …
maximal normal subgroup of	#6632111, #6668398, #6668399, …
maximal quotient of	#332842, #332848, #332855, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 23475);