#108010

order	512
nilpotency class	3
isoclinism	isoclinic class of #108006
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	C₂²⋊#95
commutator	C₄²	C₂×C₄²
Frattini, cosocle	C₂²×C₄²	C₂³
prime radical	C₂³×D₄	D₄
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

896 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 48
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 48 points:

(1,2,5,9,6,8,4,14)(3,10,13,15,12,16,11,7)(17,18)(19,27)(20,31,32,41)(21,25)(22,26)(23,29)(24,33,39,35)(28,44,46,48)(30,40)(34,38)(36,42)(37,47,43,45);(1,3)(2,7,8,15)(4,11)(5,13)(6,12)(9,10,14,16)(17,19)(18,23,26,27)(20,28)(21,34)(22,29)(24,37,39,43)(25,40,42,38)(30,36)(31,44,41,48)(32,46)(33,47)(35,45);(2,8)(4,5)(7,15)(11,13)(17,20)(18,24)(19,28)(21,35)(22,32)(23,37)(25,41)(26,39)(27,43)(29,46)(30,47)(31,42)(33,36)(34,45)(38,44)(40,48)

Polycyclic presentation:

2^9|a^2=f,b^2=g,d^2=h,e^2=i,f^2=h,b^a=bd,c^a=ce,d^a=dh,e^a=eh,g^a=ghi,d^b=di,e^b=ehi,d^c=dhi,e^c=ei,f^c=fhi

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72240822, #72257693, #72257700
maximal normal subgroup of	#72240822, #72257693, #72257700
maximal quotient of	#72240707

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15206);