#376943

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	C₂²×#41	C₂³
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1459 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)(5,6,11,16)(7,17,20,31)(8,13,19,28)(9,22,24,29)(10,25,12,15)(14,26,30,23)(18,27,34,36)(21,35,33,32);(5,7)(6,13)(8,12)(9,18)(10,19)(11,20)(14,27)(15,31)(16,28)(17,25)(21,23)(22,32)(24,34)(26,33)(29,35)(30,36);(5,8,12,20,11,19,10,7)(6,13,25,17,16,28,15,31)(9,21,23,34,24,33,26,18)(14,27,22,32,30,36,29,35);(1,3,4,2)(5,9)(6,14)(7,18)(8,21)(10,26)(11,24)(12,23)(13,27)(15,29)(16,30)(17,32)(19,33)(20,34)(22,25)(28,36)(31,35)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=hi,e^2=i,g^2=i,b^a=be,c^a=cf,d^a=de,e^a=ei,c^b=cg,e^b=ei,f^b=fi,g^b=gi,e^c=ei,f^c=fi,g^c=gi,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, 284139);