#6342510

order	512
nilpotency class	2
isoclinism	isoclinic class of D₄²
automorphism	Aut(G) order 196608
homology	Abelianization (H₁) = C₂³×C₄² · Schur multiplier (H₂) = C₂⁷×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₄×C₈	C₂⁴
commutator, socle	C₂²	C₂³×C₄²
Frattini, cosocle	C₄²	C₂⁵
prime radical	C₂²×#41	C₂×C₄
LCS	G ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₄×C₈ ◁ G
LECS	G ▷ C₄² ▷ C₂² ▷ C₁

2091 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 32
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 5 generators on 32 points:

(1,2,4,8,5,6,11,13)(3,7,10,14,9,12,15,16)(17,18)(19,22)(20,24)(21,23)(25,29)(26,28)(27,30)(31,32);(17,19,20,25,21,26,27,31)(18,22,24,29,23,28,30,32);(2,6)(7,12)(8,13)(14,16)(18,23)(22,28)(24,30)(29,32);(1,3)(2,7)(4,10)(5,9)(6,12)(8,14)(11,15)(13,16);(3,9)(7,12)(10,15)(14,16)

Polycyclic presentation:

2^9|a^2=f,b^2=g,f^2=h,g^2=i,c^a=chi,e^d=eh

Conjugacy classes & character table

200 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72322104, #72322134, #72322165
maximal normal subgroup of	#72322104, #72322134, #72322165

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6249706);