#30483

order	256
nilpotency class	3
isoclinism	isoclinic class of #30428
automorphism	Aut(G) order 4096
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

500 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

37 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#6667751, #6667759, #6667763, …
maximal normal subgroup of	#6667751, #6667759, #6667763, …
maximal quotient of	#333136, #333141, #333149, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 23471);