#376950

order	512
nilpotency class	3
isoclinism	isoclinic class of #24472
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁵×C₄

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2410
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

1471 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 36
transitive degree	μ_t(G) = 64

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

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

Polycyclic presentation:

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

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, 284146);