#353864

order	512
nilpotency class	2
isoclinism	isoclinic class of #2403
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂³	C₂²×C₄²
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2286 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 256

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

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

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=g,g^2=i,b^a=be,c^a=cf,d^b=di

Conjugacy classes & character table

152 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312905, #72312948, #72312967, …
maximal normal subgroup of	#72312905, #72312948, #72312967, …
maximal quotient of	#72321552, #72321653

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261060);