#186617

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

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

3410 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)(3,4)(5,6,8,10,9,12,7,11)(13,14)(15,24)(16,27)(17,23)(18,22)(19,21)(20,25)(26,28);(2,4)(6,10)(7,8)(11,12)(13,15,19,16)(14,20,21,28)(17,26,18,25)(22,27,23,24);(13,16,19,15)(14,20,21,28)(17,25,18,26)(22,27,23,24);(14,21)(20,28)(22,23)(24,27)

Polycyclic presentation:

2^9|a^2=g,b^2=h,c^2=h,e^2=i,f^2=h,g^2=hi,b^a=be,c^a=cf,d^a=dh,e^a=eh,f^a=fh,e^b=ei,f^b=fh,g^b=ghi,e^c=eh,f^c=fh

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310308, #72310309, #72310315
maximal normal subgroup of	#72310308, #72310309, #72310315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93813);