#369554

order	512
nilpotency class	3
isoclinism	isoclinic class of #24987
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂³×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2420
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

726 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:

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

Polycyclic presentation:

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

Conjugacy classes & character table

80 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 276750);