#358868

order	512
nilpotency class	3
isoclinism	isoclinic class of #2002
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂³	C₂×#120
prime radical	C₂×#537	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

2616 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)(14,18)(16,23)(19,27)(20,24)(21,25)(26,28);(5,6)(7,10)(8,9)(11,12)(13,14)(15,19)(16,24)(17,20)(18,23)(21,26)(22,28)(25,27);(6,9)(10,12)(13,15,17,22)(14,19,20,28)(16,25,23,21)(18,26,24,27);(5,7,8,11)(6,10,9,12)

Polycyclic presentation:

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

Conjugacy classes & character table

104 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239475, #72266618
maximal normal subgroup of	#72239475, #72266618
maximal quotient of	#72239481, #72266612

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 266064);