#209183

order	512
nilpotency class	3
isoclinism	isoclinic class of #19072
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×H(3,𝔽₄)	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₄³ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1072 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:

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

Polycyclic presentation:

2^9|a^2=g,d^2=gh,e^2=h,f^2=hi,b^a=be,c^a=cf,e^a=eh,f^a=fhi,d^b=dh,e^b=eh,f^b=fi,d^c=dhi,e^c=ei,f^c=fhi

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, 116379);