#353119

order	512
nilpotency class	2
isoclinism	isoclinic class of #534
automorphism	?
homology	Abelianization (H₁) = C₂×C₄³ · Schur multiplier (H₂) = C₂⁵×C₄²

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, Frattini, cosocle	C₂³×C₄	C₂⁴
commutator	C₂²	C₂×C₄³
socle	C₂⁴	C₂³×C₄
prime radical	C₂⁵	C₂²×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2682 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)(13,14,16,21,18,20,17,22)(15,19,24,27,23,26,25,28);(2,4)(9,10,11,12)(15,23)(19,26)(24,25)(27,28);(13,15,17,25,18,23,16,24)(14,19,22,28,20,26,21,27);(14,20)(19,26)(21,22)(27,28)

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72213009, #72215773, #72215932, …
maximal normal subgroup of	#72213009, #72215773, #72215932, …
maximal quotient of	#72212990, #72321669

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260315);