#353291

order	512
nilpotency class	2
isoclinism	isoclinic class of #2846
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁹

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	C₂⁶	C₂×C₄
LCS	G ▷ C₂³ ▷ C₁
UCS	C₁ ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

5258 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

176 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72308505, #72308525, #72308551, …
maximal normal subgroup of	#72308505, #72308525, #72308551, …
maximal quotient of	#72321588, #72321933, #72321942, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 260487);