#357499

order	512
nilpotency class	3
isoclinism	isoclinic class of #23391
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁹

Aliases

No aliases for this group.

Subgroups

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

5154 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)(9,10,11,12)(13,14)(15,23)(16,22)(17,20)(18,21)(19,24)(25,28)(26,27);(2,4)(5,6)(7,8)(13,15,18,26)(14,19,21,28)(16,25,17,24)(20,27,22,23);(6,8)(14,20)(16,17)(19,27)(21,22)(23,28)(24,25);(14,21)(19,28)(20,22)(23,27)

Polycyclic presentation:

2^9|a^2=h,b^2=i,e^2=i,f^2=i,b^a=be,c^a=cf,d^a=di,e^a=ei,f^a=fi,c^b=cg,e^b=ei,f^b=fi,e^c=ei,f^c=fi

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315779, #72315786, #72315818, …
maximal normal subgroup of	#72315779, #72315786, #72315818, …
maximal quotient of	#72317054

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 264695);