#15483

order	256
nilpotency class	2
isoclinism	isoclinic class of #15257
automorphism	Aut(G) order 65536
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, commutator, Frattini, socle, cosocle	C₂⁴	C₂⁴
prime radical	C₂⁵	C₂³
LCS	G ▷ C₂⁴ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ G
LECS	G ▷ C₂⁴ ▷ C₁

693 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

2^8|a^2=fg,b^2=ef,c^2=e,b^a=be,c^a=cf,d^a=dh,c^b=cg,d^b=de

Conjugacy classes & character table

58 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₂×#15483, #609067, #609202, …
maximal normal subgroup of	C₂×#15483, #609067, #609202, …
maximal quotient of	#154679, #154834, #154864, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(256, 8471);