#392944

order	512
nilpotency class	3
isoclinism	isoclinic class of #32950
automorphism	?
homology	Abelianization (H₁) = C₂³×C₄ · Schur multiplier (H₂) = C₂⁶

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂²	#2847
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

1368 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)(6,12)(7,16)(8,20)(9,10)(13,26)(15,23)(18,19)(21,22)(25,28)(27,34)(30,31)(35,36);(1,3,4,2)(5,6)(7,13)(8,14)(9,23)(10,12)(11,15)(16,24)(17,27)(18,33)(19,26)(20,28)(21,25)(22,29)(30,34)(31,36)(32,35);(5,7,10,16,11,19,9,18)(6,13,12,24,15,26,23,33)(8,17,21,30,22,32,20,31)(14,27,25,34,29,35,28,36);(5,8)(6,14)(7,17)(9,20)(10,21)(11,22)(12,25)(13,27)(15,29)(16,30)(18,31)(19,32)(23,28)(24,34)(26,35)(33,36)

Polycyclic presentation:

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

Conjugacy classes & character table

92 conjugacy classes — view the full character table.

Relationships

maximal subgroup of	#72312202, #72312265, #72312295
maximal normal subgroup of	#72312202, #72312265, #72312295

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 300140);