#376948

order	512
nilpotency class	3
isoclinism	isoclinic class of #24408
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×#2008	C₂
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂² ◁ C₂³×C₄ ◁ G
LECS	G ▷ C₂³×C₄ ▷ C₂ ▷ C₁

1539 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)(8,21,25,29)(9,14,10,15)(12,26,28,19)(13,23,30,24)(18,27,22,36)(32,34,35,33);(5,7)(6,12)(8,22)(9,26)(10,19)(11,20)(13,27)(14,17)(15,31)(16,28)(18,25)(21,35)(23,33)(24,34)(29,32)(30,36);(5,8)(6,13)(7,18,20,22)(9,24)(10,23)(11,25)(12,27,28,36)(14,29)(15,21)(16,30)(17,32,31,35)(19,34,26,33);(1,3,4,2)(6,14)(9,10)(12,17)(13,29)(15,16)(19,26)(21,30)(23,24)(27,32)(28,31)(33,34)(35,36)

Polycyclic presentation:

2^9|a^2=h,c^2=ef,d^2=hi,e^2=i,f^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,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.

Relationships

maximal subgroup of	#72280941, #72281002, #72281232
maximal normal subgroup of	#72280941, #72281002, #72281232
maximal quotient of	#72313421

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 284144);