#6386108

order	512
nilpotency class	3
isoclinism	isoclinic class of #6342764
automorphism	?
homology	Abelianization (H₁) = C₂⁴×C₄ · Schur multiplier (H₂) = C₂⁸×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂³×C₂²⋊C₄
commutator	C₂³	C₂⁴×C₄
Frattini, cosocle	C₂⁴	C₂⁵
prime radical	C₂²×#120	C₂²
LCS	G ▷ C₂³ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₂²⋊C₄ ◁ G
LECS	G ▷ C₂⁴ ▷ C₂ ▷ C₁

3744 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

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

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

Polycyclic presentation:

2^9|a^2=h,b^2=i,c^2=f,b^a=bf,c^a=cg,e^a=ef,g^a=gi,d^b=df,h^c=hi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72310693
maximal normal subgroup of	#72310693

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6293304);