#357952

order	512
nilpotency class	3
isoclinism	isoclinic class of #23387
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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	C₂³×#41	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂⁴ ◁ C₂²×#117 ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2822 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)(9,10,11,12)(14,19)(15,23)(16,17)(20,28)(21,22)(25,26);(5,6)(7,8)(13,14)(15,24)(16,22)(17,19)(18,21)(20,25)(23,28)(26,27);(2,4)(6,8)(13,15,16,23,18,26,17,25)(14,20,19,27,21,28,22,24);(14,21)(19,22)(20,28)(24,27)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72315984, #72315987, #72316014
maximal normal subgroup of	#72315984, #72315987, #72316014

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 265148);