#107812

order	512
nilpotency class	3
isoclinism	isoclinic class of #107810
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	#3444	C₂²
LCS	G ▷ C₄² ▷ C₂² ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂²×C₄² ◁ G
LECS	G ▷ C₂²×C₄² ▷ C₂² ▷ C₁

728 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

62 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72286729, #72286788, #72286836
maximal normal subgroup of	#72286729, #72286788, #72286836

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 15008);