#383984

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2688
commutator	C₂²×C₄	C₂³×C₄
Frattini, cosocle	C₂³×C₄	C₂⁴
prime radical	C₂⁴	C₂×#41
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

805 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)(7,16,19,12)(8,21,24,26)(9,23,13,29)(10,27,30,14)(17,33,22,35)(18,34,25,36)(20,31,28,32);(5,7,10,28,11,19,30,20)(6,12,14,32,15,16,27,31)(8,22,23,34,24,17,29,36)(9,25,26,35,13,18,21,33);(5,8)(6,13)(7,17,19,22)(9,15)(10,29)(11,24)(12,18,16,25)(14,26)(20,36,28,34)(21,27)(23,30)(31,33,32,35);(5,9,11,13)(6,8,15,24)(7,18,19,25)(10,21,30,26)(12,17,16,22)(14,29,27,23)(20,33,28,35)(31,36,32,34)

Polycyclic presentation:

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

Conjugacy classes & character table

68 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 291180);