#354245

order	512
nilpotency class	3
isoclinism	isoclinic class of #3019
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×AGL(1,ℤ/8)	C₂²
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2088 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 24
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 24 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	C₄×#21473
maximal normal subgroup of	C₄×#21473
maximal quotient of	#72279328, #72279584

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 261441);