#147419

order	512
nilpotency class	4
isoclinism	isoclinic class of #13068
automorphism	?
homology	Abelianization (H₁) = C₂×C₄² · Schur multiplier (H₂) = C₂⁴×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1893
commutator	C₂×C₈	C₂×C₄²
Frattini, cosocle	C₂³×C₈	C₂³
prime radical	C₂²×#135	C₂²
LCS	G ▷ C₂×C₈ ▷ C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ C₂×#405 ◁ G
LECS	G ▷ C₂³×C₈ ▷ C₄ ▷ C₂ ▷ C₁

714 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

A minimal faithful permutation representation — 3 generators on 36 points:

(2,4)(5,6,11,18,13,20,10,17)(7,21,25,34,26,36,24,27)(8,23,14,15,29,31,28,35)(9,30,32,22,33,19,12,16);(5,7,12,15,10,24,33,23,13,26,32,35,11,25,9,31)(6,14,19,34,18,29,16,36,20,28,30,27,17,8,22,21);(1,2,3,4)(5,8)(6,15)(7,22)(9,27)(10,14)(11,28)(12,21)(13,29)(16,26)(17,31)(18,23)(19,24)(20,35)(25,30)(32,36)(33,34)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72239801, #72239804, #72239819, …
maximal normal subgroup of	#72239801, #72239804, #72239819, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 54615);