#186616

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	C₂²×#550	C₂
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

3706 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)(13,14,17,21,18,19,16,25)(15,22,24,20,23,28,26,27);(2,4)(6,10)(7,8)(11,12)(14,19)(16,17)(20,27)(24,26);(13,15,18,23)(14,20,19,27)(16,26,17,24)(21,28,25,22);(14,19)(20,27)(21,25)(22,28)

Polycyclic presentation:

2^9|a^2=g,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	#72310306, #72310307, #72310315
maximal normal subgroup of	#72310306, #72310307, #72310315

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 93812);