#147816

order	512
nilpotency class	4
isoclinism	isoclinic class of #129633
automorphism	?
homology	Abelianization (H₁) = C₂²×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#1790
commutator	C₂²×D₄	C₂²×C₄
Frattini, cosocle	C₂³×D₄	C₂³
prime radical	C₂×#534	C₂²
LCS	G ▷ C₂²×D₄ ▷ C₂² ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂⁴ ◁ C₂³×D₄ ◁ G
LECS	G ▷ C₂³×D₄ ▷ C₂² ▷ C₂ ▷ C₁

1201 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

56 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72280770, #72303129, #72303135, …
maximal normal subgroup of	#72280770, #72303129, #72303135, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 55012);