#386379

order	512
nilpotency class	3
isoclinism	isoclinic class of #31418
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁴×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	C₂×#575	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1006 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)(6,11)(8,20)(9,16)(12,28)(14,24)(15,26)(17,32)(18,25)(21,27)(22,23)(29,31)(34,35);(5,6)(7,15)(8,21)(9,24)(10,14)(11,16)(12,19)(13,22)(17,33)(18,28)(20,30)(23,27)(25,26)(29,35)(31,32)(34,36);(5,7,16,18,10,19,9,25)(6,12,11,26,14,15,24,28)(8,17,22,35,23,34,20,32)(13,29,27,36,30,31,21,33);(5,8)(6,13)(7,17)(9,20)(10,23)(11,27)(12,29)(14,30)(15,31)(16,22)(18,35)(19,34)(21,24)(25,32)(26,36)(28,33)

Polycyclic presentation:

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

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, 293575);