#353654

order	512
nilpotency class	3
isoclinism	isoclinic class of #468
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂³×C₄	C₂×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂⁴	C₂²×D₄
prime radical	C₂⁵	C₂×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ C₂³×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2394 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)(6,8)(9,10,11,12)(13,14)(15,25)(16,28)(17,22)(18,23)(19,24)(20,26)(21,27);(2,4)(13,15,18,16,19,27,17,26)(14,20,22,21,24,28,23,25);(5,6,7,8)(13,16,18,27,19,26,17,15)(14,21,22,28,24,25,23,20);(13,17,19,18)(14,22,24,23)(15,26,27,16)(20,21,28,25)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=f,d^2=i,e^2=i,f^2=i,h^2=i,b^a=be,c^a=cf,e^a=ei,f^a=fi,h^a=hi,c^b=ci

Conjugacy classes & character table

128 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72211371, #72213009, #72213456, …
maximal normal subgroup of	#72211371, #72213009, #72213456, …
maximal quotient of	#72211357, #72322995

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260850);