#194013

order	512
nilpotency class	3
isoclinism	isoclinic class of #18524
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

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	C₂²×#41	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1406 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)(5,6,9,11,10,7,8,12)(13,14)(15,21,23,28)(16,18)(17,20)(19,22,26,25)(24,27);(1,2,3,4)(5,7,10,6)(8,12,9,11)(14,18)(17,24)(19,26)(22,25);(13,15,16,23)(14,19,18,26)(17,25,24,22)(20,28,27,21);(5,8,10,9)(6,11,7,12)

Polycyclic presentation:

2^9|a^2=g,b^2=eghi,c^2=i,d^2=h,e^2=h,f^2=i,g^2=h,b^a=be,c^a=cf,d^a=dh,e^b=eh,f^b=fi,g^b=gh,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, 101209);