#193205

order	512
nilpotency class	3
isoclinism	isoclinic class of #18746
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×#532	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂³ ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

1136 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

86 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72279276, #72309525, #72309597
maximal normal subgroup of	#72279276, #72309525, #72309597
maximal quotient of	#72279385

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 100401);