#414645

order	512
nilpotency class	3
isoclinism	isoclinic class of #393963
automorphism	?
homology	Abelianization (H₁) = C₂⁴ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator, Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂²×#127	C₂²
LCS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1964 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 28
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 4 generators on 28 points:

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

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240389, #72240393, #72261364, …
maximal normal subgroup of	#72240389, #72240393, #72261364, …
maximal quotient of	#72257813

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 321841);