#108357

order	512
nilpotency class	3
isoclinism	isoclinic class of #108352
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁵

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₂²×C₄	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂×#121	C₂×C₄
LCS	G ▷ C₂²×C₄ ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

761 conjugacy classes of subgroups.

Full subgroup data →

Presentation

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

A minimal faithful permutation representation — 3 generators on 32 points:

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

Polycyclic presentation:

2^9|a^2=f,b^2=g,e^2=i,g^2=hi,b^a=bd,c^a=ce,d^a=dh,e^a=ei,g^a=gh,d^b=dh,e^b=eh,f^b=fh,d^c=dh,e^c=ei

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15553);