#183497

order	512
nilpotency class	3
isoclinism	isoclinic class of #3264
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂²×C₄	C₂²×D₄
commutator	C₂×C₄	C₂²×C₄²
Frattini, cosocle	C₂×C₄²	C₂⁴
socle	C₂³	C₂×C₄×D₄
prime radical	C₂⁴	C₄×D₄
LCS	G ▷ C₂×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂²×C₄ ◁ C₂×C₄×M₁₆ ◁ G
LECS	G ▷ C₂×C₄² ▷ C₂² ▷ C₁

852 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:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,c^2=ef,e^2=h,f^2=i,b^a=be,c^a=ch,e^b=eh,f^b=fh,d^c=dhi

Conjugacy classes & character table

116 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72343941, #72343957, #72343963
maximal normal subgroup of	#72343941, #72343957, #72343963

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 90693);