#194825

order	512
nilpotency class	3
isoclinism	isoclinic class of #18252
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂×#120
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂×#2540	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂² ▷ C₁

1812 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 102021);