#392608

order	512
nilpotency class	3
isoclinism	isoclinic class of #32898
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁸

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center	C₂×C₄	C₂×#120
commutator	C₂⁴	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
socle	C₂²	C₄×#120
prime radical	C₂×#585	C₂²
LCS	G ▷ C₂⁴ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂×C₄ ◁ C₂³×C₈ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

3403 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 128

A minimal faithful permutation representation — 4 generators on 36 points:

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

Polycyclic presentation:

2^9|a^2=h,b^2=h,h^2=i,b^a=be,c^a=cf,d^a=dg,f^b=fi,e^c=ei,g^c=gi,f^d=fi

Conjugacy classes & character table

98 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72250134, #72250285, #72257156, …
maximal normal subgroup of	#72250134, #72250285, #72257156, …
maximal quotient of	#72285816, #72287524, #72317360, …

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 299804);