#194129

order	512
nilpotency class	3
isoclinism	isoclinic class of #18562
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	C₄²	C₂³×C₄
Frattini, cosocle	C₂×C₄²	C₂⁴
prime radical	C₂×#3251	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1294 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:

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

Polycyclic presentation:

2^9|a^2=g,b^2=egh,d^2=hi,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,d^b=dh,e^b=eh,f^b=fi,g^b=gh,d^c=dh,e^c=ei,f^c=fi,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, 101325);